Nanomagnetism - Institut...

Lecture notes on

Nanomagnetism

Olivier Fruchart

Institut Neel (CNRS & UJF) – Grenoble

Version : December 2, 2011

Olivier.Fruchart-at-grenoble.cnrs.fr

http://perso.neel.cnrs.fr/olivier.fruchart/

Contents

Introduction 5Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Formatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

I Setting the ground for nanomagnetism 71 Magnetic fields and magnetic materials . . . . . . . . . . . . . . . . . 7

1.1 Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Magnetic materials . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Magnetic materials under field – The hysteresis loop . . . . . 111.4 Domains and domain walls . . . . . . . . . . . . . . . . . . . . 15

2 Units in Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 The various types of magnetic energy . . . . . . . . . . . . . . . . . . 17

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Zeeman energy . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Magnetic anisotropy energy . . . . . . . . . . . . . . . . . . . 183.4 Exchange energy . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Magnetostatic energy . . . . . . . . . . . . . . . . . . . . . . . 203.6 Characteristic quantities . . . . . . . . . . . . . . . . . . . . . 20

4 Handling dipolar interactions . . . . . . . . . . . . . . . . . . . . . . 214.1 Simple views on dipolar interactions . . . . . . . . . . . . . . 214.2 Ways to handle dipolar fields . . . . . . . . . . . . . . . . . . 224.3 Demagnetizing factors . . . . . . . . . . . . . . . . . . . . . . 23

5 The Bloch domain wall . . . . . . . . . . . . . . . . . . . . . . . . . . 265.1 Simple variational model . . . . . . . . . . . . . . . . . . . . . 265.2 Exact model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Defining the width of a domain wall . . . . . . . . . . . . . . . 28

6 Magnetometry and magnetic imaging . . . . . . . . . . . . . . . . . . 296.1 Extraction magnetometers . . . . . . . . . . . . . . . . . . . . 306.2 Faraday and Kerr effects . . . . . . . . . . . . . . . . . . . . . 306.3 X-ray Magnetic Dichroism techniques . . . . . . . . . . . . . . 306.4 Near-field microscopies . . . . . . . . . . . . . . . . . . . . . . 306.5 Electron microscopies . . . . . . . . . . . . . . . . . . . . . . . 31

Problems for Chapter I 321. More about units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.1. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.2. Expressing dimensions . . . . . . . . . . . . . . . . . . . . . . . . 321.3. Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2

Contents 3

2. More about the Bloch domain wall . . . . . . . . . . . . . . . . . . . . . 332.1. Euler-Lagrange equation . . . . . . . . . . . . . . . . . . . . . . . 332.2. Micromagnetic Euler equation . . . . . . . . . . . . . . . . . . . 342.3. The Bloch domain wall . . . . . . . . . . . . . . . . . . . . . . . 34

3. Extraction and vibration magnetometer . . . . . . . . . . . . . . . . . . 363.1. Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2. Flux in a single coil . . . . . . . . . . . . . . . . . . . . . . . . . 363.3. Vibrating in a single coil . . . . . . . . . . . . . . . . . . . . . . . 363.4. Noise in the signal . . . . . . . . . . . . . . . . . . . . . . . . . . 373.5. Winding in opposition . . . . . . . . . . . . . . . . . . . . . . . . 37

4. Magnetic force microscopy . . . . . . . . . . . . . . . . . . . . . . . . . 374.1. The mechanical oscillator . . . . . . . . . . . . . . . . . . . . . . 374.2. AFM in the static and dynamic modes . . . . . . . . . . . . . . . 384.3. Modeling forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

II Magnetism and magnetic domains in low dimensions 391 Magnetic ordering in low dimensions . . . . . . . . . . . . . . . . . . 39

1.1 Ordering temperature . . . . . . . . . . . . . . . . . . . . . . 391.2 Ground-state magnetic moment . . . . . . . . . . . . . . . . . 41

2 Magnetic anisotropy in low dimensions . . . . . . . . . . . . . . . . . 422.1 Dipolar anisotropy . . . . . . . . . . . . . . . . . . . . . . . . 422.2 Projection of magnetocrystalline anisotropy due to dipolar en-

ergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.3 Interface magnetic anisotropy . . . . . . . . . . . . . . . . . . 442.4 Magnetoelastic anisotropy . . . . . . . . . . . . . . . . . . . . 46

3 Domains and domain walls in thin films . . . . . . . . . . . . . . . . . 483.1 Bloch versus Neel domain walls . . . . . . . . . . . . . . . . . 483.2 Domain wall angle . . . . . . . . . . . . . . . . . . . . . . . . 493.3 Composite domain walls . . . . . . . . . . . . . . . . . . . . . 503.4 Vortices and antivortex . . . . . . . . . . . . . . . . . . . . . . 513.5 Films with an out-of-plane anisotropy . . . . . . . . . . . . . . 52

4 Domains and domain walls in nanostructures . . . . . . . . . . . . . . 544.1 Domains in nanostructures with in-plane magnetization . . . . 544.2 Domains in nanostructures with out-of-plane magnetization . 554.3 The critical single-domain size . . . . . . . . . . . . . . . . . . 564.4 Near-single-domain . . . . . . . . . . . . . . . . . . . . . . . . 574.5 Domain walls in stripes and wires . . . . . . . . . . . . . . . . 58

5 An overview of characteristic quantities . . . . . . . . . . . . . . . . . 595.1 Energy scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 Length scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3 Dimensionless ratios . . . . . . . . . . . . . . . . . . . . . . . 61

Problems for Chapter II 621. Short exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622. Demagnetizing field in a stripe . . . . . . . . . . . . . . . . . . . . . . . 62

2.1. Deriving the field . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.2. Numerical evaluation and plotting . . . . . . . . . . . . . . . . . 63

IIIMagnetization reversal 64

4 Contents

1 Coherent rotation of magnetization . . . . . . . . . . . . . . . . . . . 641.1 The Stoner-Wohlfarth model . . . . . . . . . . . . . . . . . . . 641.2 Dynamic coercivity and temperature effects . . . . . . . . . . 65

2 Magnetization reversal in nanostructures . . . . . . . . . . . . . . . . 652.1 Multidomains under field (soft materials) . . . . . . . . . . . . 652.2 Nearly single domains . . . . . . . . . . . . . . . . . . . . . . 652.3 Domain walls and vortices . . . . . . . . . . . . . . . . . . . . 65

3 Magnetization reversal in extended systems . . . . . . . . . . . . . . . 653.1 Nucleation and propagation . . . . . . . . . . . . . . . . . . . 653.2 Ensembles of grains . . . . . . . . . . . . . . . . . . . . . . . . 65

4 What do we learn from hysteresis loops? . . . . . . . . . . . . . . . . 664.1 Magnetic anisotropy . . . . . . . . . . . . . . . . . . . . . . . 664.2 Nucleation versus propagation . . . . . . . . . . . . . . . . . . 664.3 Distribution and interactions . . . . . . . . . . . . . . . . . . . 66

Problems for Chapter III 671. Short exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672. A model of pinning - Kondorski’s law for coercivity . . . . . . . . . . . . 67

IV Precessional dynamics of magnetization 691 Ferromagnetic resonance and Landau-Lifshitz-Gilbert equation . . . . 692 Precessional switching of macrospins driven by magnetic fields . . . . 693 Precessional switching driven by spin transfer torques . . . . . . . . . 694 Precessional dynamics of domain walls and vortices – Field and current 69

Problems for Chapter IV 70

V Magnetic heterostructures: from specific properties to applications 711 Coupling effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 Magnetotransport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 Integration for applications . . . . . . . . . . . . . . . . . . . . . . . . 71

Problems for Chapter V 72

Appendices 73Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Bibliography 75

Introduction

Content

This manuscript is based on series of lectures about Nanomagnetism. Parts havebeen given at the European School on Magnetism, at the Ecole Doctorale de Physiquede Grenoble, or in Master lectures at the Cadi Ayyad University in Marrakech.

Nanomagnetism may be defined as the branch of magnetism dealing with low-dimension systems and/or systems with small dimensions. Such systems may displaybehaviors different from those in the bulk, pertaining to magnetic ordering, mag-netic domains, magnetization reversal etc. These notes are mainly devoted to theseaspects, with an emphasis on magnetic domains and magnetization reversal.

Spintronics, i.e. the physics linking magnetism and electrical transport such asmagnetoresistance, is only partly and phenomenologically mentioned here. We willconsider those cases where spin-polarized currents influence magnetism, however notwhen magnetism influences the electronic transport.

This manuscript is only an introduction to Nanomagnetism, and also stickingto a classical and phenomenological descriptions of magnetism. It targets beginnersin the field, who need to use basics of Nanomagnetism in their research. Thus theexplanations aim at remaining understandable by a large scope of physicists, whilestaying close to the state-of-the art for the most advanced or recent topics.

Finally, these notes are never intended to be in a final form, and are thusby nature imperfect. The reader should not hesitate to report errors ormake suggestions about topics to improve or extend further. A consequenceis that it is probably unwise to print this document. Its use as an electronic file isanyhow preferable to benefit from the included links within the file. At present onlychapters I and II are more or less completed.

Notations

As a general rule, the following typographic rules will be applied to variables:

Characters

• A microscopic extensive or intensive quantity appears as slanted uppercase orGreek letter, such as H for the magnitude of magnetic field, E for a densityof energy expressed in J/m3, ρ for a density.

• An extensive quantity integrated over an entire system appears as handwrittenuppercase. A density of energy E integrated over space will thus be writtenE , and expressed in J.

5

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6 Introduction

• A microscopic quantity expressed in a dimensionless system appears as a hand-written lowercase, such as e for an energy or h for a magnetic field normalizedto a reference value. Greek letters will be used for dimensionless versions ofintegrated quantities, such as ε for a total energy.

• Lengths and angles will appear as lower case roman or Greek letters, such asx for a length or α for an angle. If needed, a specific notation is introducedfor dimensionless lengths.

• A vector appears as bold upright, with no arrow. Vectors may be lowercase,uppercase, handwritten or Greek, consistently with the above rules. We willthus write H for a magnetic field, h its dimensionless counterpart, k a unitvector along direction z, M or µ a magnetic moment.

Mathematics

• The cross product of two vectors A and B will be written A×B.

• The following shortcut may be used: ∂xθ for ∂θ/∂x, or ∂nxnθ for ∂nθ∂xn

.

• The elementary integration volume integration may be written d3r or dτ , andthe surface on: d2r or dS.

Units

• The International system of units (SI) will be used for numerical values.

• B will be called magnetic induction, H magnetic field, and M magnetization.We will often use the name magnetic field in place of B when no confusionexists, i.e. in the absence of magnetization (in vacuum). This is a shortcutfor B/µ0, to be expressed in Teslas.

Special formatting

Special formatting is used to draw the attention of the reader at certains aspects,as illustrated below.

Words highlighted like this are of special importance, either in the local context,or when they are important concepts introduced for the first time.

The hand sign will be associated with hand-wavy arguments and take-away messages.

The slippery sign will be associated with misleading aspects and fine points.

Chapter I

Setting the ground fornanomagnetism

Overview

A thorough introduction to Magnetism[1, 2, 3] and Micromagnetismand Nanomagnetism[4, 5, 6, 7] may be sought in dedicated books. Thischapter only serves as an introduction to the lecture, and it is not com-prehensive. We only provide general reminders about magnetism, micro-magnetism, and of some characterization techniques useful for magneticfilms and nanostructures.

1 Magnetic fields and magnetic materials

1.1 Magnetic fields

Electromagnetism is described by the four Maxwell equations. Let us consider thesimple case of stationary equations. Magnetic induction B then obeys two equations:

curl B = µ0 j (I.1)

div B = 0 (I.2)

j being a volume density of electrical current. j appears as a source of inductionloops, similar to electrostatics where the density of electric charge ρ is the sourceof radial electric field E. Let us first consider the simplest case for an electric cur-rent, that of an infinite linear wire with total current I. We shall use cylindricalcoordinates. Any plane comprising the wire is a symmetry element for the currentand thus an antisymmetric element for the induced induction (see above equations),which thus is purely orthoradial and described by the component Bθ only. In addi-tion the system is invariant by rotation around and translation along the wire, sothat Bθ depends neither on θ nor z, however solely on the distance r to the wire.Applying Stokes theorem to an orthoradial loop with radius r (Figure I.1) readilyleads to:

Bθ(r) =µ0I

2πr(I.3)

7

8 Chapter I. Setting the ground for nanomagnetism

I

r uθ

B=B(r)uθ θ

Figure I.1: So-called Oersted magnetic inductionB, arising from an infinite andlinear wire with an electrical current I.

This is the so-called Oersted induction or Oersted field, named after its discov-ery in 1820 by Hans-Christian Oersted. This discovery was the first evidence ofthe connection of electricity and magnetism, and is therefore a foundation for thedevelopment of electromagnetism. Notice the variation with 1/r. Let us consideran order of magnitude for daily life figures. For I = 1 A and r = 10−2 m we findB = 2× 10−5 T. This magnitude is comparable to the earth magnetic field, around50µT. It is weak compared to fields arising from permanent magnets or dedicatedelectromagnets and superconducting magnets.

We may argue that there exists no infinite line of current. The Biot and Savartlaw describes instead the elementary contribution to induction δB at point P, arisingfrom an elementary part of wire δ` at point Q with a current I:

δB(P ) =µ0I δ`×QP

4πQP 3(I.4)

Notice this time the variation as 1/r2. This can be understood qualitativelyas a macroscopic (infinite) line is the addition (mathematically, the integral) ofelementary segments, and we have

∫1/r2 = 1/r. It may also be argued that there

exists no elementary segments of current for conducting wirings, however only closedcircuits (loops), with a uniform current I along its length. When viewed as a distancefar compared to its dimensions, a loop of current may be considered as a pinpointmagnetic dipole µ. This object is an example of a magnetic moment. For a planarloop µ = IS where S is the surface vector normal to the plane of the loop, orientedaccordingly with the electrical current. Here it appears clearly that the SI unit fora magnetic moment is A.m2. The expansion of the Biot and Savart law leads to theinduction arising from a dipole at long distance r:

B(r) =µ0

4πr3

[3

(µ.r)r

r2− µ

]. (I.5)

Let us note now the variation with 1/r3. This may be understood as the firstderivative of the variation like 1/r2 arising from an elementary segment, due tonearby regions run by opposite vectorial currents j (e.g. the opposite parts of aloop).

Table I.1 summarizes the three cases described above.

I.1. Magnetic fields and magnetic materials 9

Table I.1: Long-distance decay of induction arising from various types of currentdistributions

Case DecayInfinite line of current 1/r

Elementary segment 1/r2

Current loop (magnetic dipole) 1/r3

Table I.2: Main features of a few important ferromagnetic materials: Ordering(Curie) temperature TC, spontaneous magnetization Ms and a magnetocristallineanisotropy constant K at 300 K (The symmetry of the materials, and hence theorder of the anisotropy constants provided, is not discussed here)

Material TC (K) Ms (kA/m) µ0Ms (T) K (kJ/m3)

Fe 1043 1730 2.174 48Co 1394 1420 1.784 530Ni 631 490 0.616 -4.5

Fe304 858 480 0.603 -13BaFe12O19 723 382 0.480 250Nd2Fe14B 585 1280 1.608 4900

SmCo5 995 907 1.140 17000Sm2Co17 1190 995 1.250 3300FePt L10 750 1140 1.433 6600CoPt L10 840 796 1.000 4900

Co3Pt 1100 1100 1.382 2000

1.2 Magnetic materials

A magnetic material is a body which displays a magnetization M(r), i.e. a volumedensity of magnetic moments. The SI unit for magnetization therefore appears nat-urally A.m2/m3, thus A/mI.1. In any material some magnetization may be inducedunder the application of an external magnetic field H. We define the magnetic sus-ceptibility χ with M = χH. This polarization phenomenon is named diamagnetismfor χ < 0 and paramagnetism for χ > 0.

Diamagnetism arises from a Lenz-like law at the microscopic level (electronicorbitals), and is present in all materials. χdia is constant with temperature and itsvalue is material-dependent, however roughly of the order of 10−5. Peak values arefound for Bi (χ = −1.66 × 10−4) and graphite along the c axis (χ ≈ −4 × 10−4).Such peculiarities may be explained by the low effective mass of the charge carriers

I.1We shall always use strictly the names magnetic moment and magnetization. Experimen-tally some techniques provide direct or indirect access to magnetic moments (e.g. an extractionmagnetometer, a SQUID, magnetic force microscopy), other provide a m ore natural access to mag-netization, often through data analysis (e.g. magnetic dichroism of X-rays, electronic or nuclearresonance).


involved.Paramagnetism arises from partially-filled orbitals, either forming bands or lo-

calized. The former case is called Pauli paramagnetism. χ is then temperature-independent and rather weak, again of the order of 10−5. The later case is calledCurie paramagnetism, and χ scales with 1/T . A useful order of magnitude in Curieparamagnetism to keep in mind is that a moment of 1µB gets polarized at 1 K underan induction of 1 T.

Only certain materials give rise to paramagnetism, in particular metals or in-sulators with localized moments. Then diamagnetism and paramagnetism add up,which may give give to an overall paramagnetic of diamagnetic response.

Finally, in certain materials microscopic magnetic moments are coupled througha so-called exchange interaction, leading to the phenomenon of magnetic orderingat finite temperature and zero field. For a first approach magnetic ordering may bedescribed in mean field theory modeling a molecular field, as we will detail for lowdimension systems in Chapter II. The main types of magnetic ordering are:

• Ferromagnetism, characterized by a positive exchange interaction, end favor-ing the parallel alignement of microscopic moments. This results in the oc-currence of a spontaneous magnetization Ms

I.2. In common cases Ms is of theorder of 106 A/m, which is very large compared to magnetization arising fromparamagnetism or diamagnetism. The ordering occurs only at and below atemperature called the Curie temperature, written TC. The only three pureelements ferromagnetic at room temperature are the 3d metals Fe, Ni andCo (Table I.2).

• Antiferromagnetism results from a negative exchange energy, favoring the an-tiparallel alignement of neighboring momentsI.3 leading to a zero net magneti-zation Ms at the macroscopic scale. The ordering temperature is in that casecalled the Neel temperature, and is written TN.

• Ferrimagnetism arises in the case of negative exchange coupling between mo-ments of different magnitude, because located each on a different sublatticeI.4,leading to a non-zero net magnetization. The ordering temperature is againcalled Curie temperature.

Let us consider the simple case of a body with uniform magnetization, for ex-ample a spontaneous magnetization Ms = Msk (Figure I.2). It is readily seen thatthe equivalent current loops modeling the microscopic moments cancel each otherfor neighboring loops: only currents at the perimeter remain. The body may thusbe modeled as a volume whose surface carries an areal density of electrical current,whose magnitude projected along k is Ms. This highlights a practical interpretationof the magnitude of magnetization expressed in A/m.

Let us stress a fundamental quantitative difference with Oersted fields. Weconsider again a metallic wire carrying a current of 1 A. For a cross-section of1 mm2 a single wiring has 1000 turns/m. The equivalent magnetization would be103 A/m, which is three orders of magnitude smaller than Ms of usual ferromagnetic

I.2The s in Ms is confusing between the meanings of spontaneous and saturation. We will discussthis fine point in the next paragraph

I.3More complexe arrangements, non-colinear like spiraling, exist like in the case of CrI.4Similarly to antiferromagnetism, more complex arrangements may be found


Figure I.2: Amperian description of a ferro (or ferri-)magnetic material: microscopiccurrents cancel each other between neighboring regions, except at the perimeter ofthe body.

materials. Thus a significant induction may easily be obtained from the stray field ofa permanent magnet, of the order of µ0Ms ≈ 1 T. It is possible to reach magnitudeof induction of several Teslas with wirings, however with special designs: largeand thick water-cooled coils to increase the current density and total value, or usesuperconducting wires however requiring their use at low temperature, or use pulsedcurrents with high values, this time requiring small dimensions to minimize self-inductance.

Let us finally recall the relationship between induction, magnetic field and mag-netization:

B = µ0(H + M) (I.6)

This relationship may be proven starting from Maxwell’s equations, consideringas two different entities the free electrical charges, and the so-called bound chargesgiving rise to the magnetization M.

1.3 Magnetic materials under field – The hysteresis loop

Let us consider a system mechanically fixed in space, subjected to an applied mag-netic field H. This field gives rise to a Zeeman energy, written EZ = −µ0Ms.Hfor a volume density, or EZ = −µ0µ.H for the energy of a magnetic moment. Theconsequence is that magnetization will tend to align itself along H, which shall beattained for a sufficient magnitude of H. This process is called a magnetizationprocess, or magnetization reversal. The quantity considered or measured may bea moment or magnetization, the former in magnetometers and the latter in somemagnetic microscopes or in the Extraordinary Hall Effect, for example. It is oftendisplayed, in models or as the result of measurements, as a hysteresis loop, alsocalled magnetization loop or magnetization curve. The horizontal axis is often H orµ0H, while the y axis is the projection of the considered quantity along the directionof H [e.g.: (M.H)/H].


Ms

Mr

Hc

Recoil loop

Firstloop

H

MdM

H

(a) (b)

Figure I.3: (a) Typical hysteresis loop illustrating the definition of coercivity Hc,saturation Ms and remanent Mr magnetization. A minor (recoil) loop as well as afirst magnetization loop are shown in thinner lines (b) The losses during a hysteresisloop equal the area of the loop.

Hysteresis loops are the most straightforward and widespread characterizationof magnetic materials. We will thus discuss it in some details, thereby introducingimportant concepts for magnetic materials and their applications. We restrict thediscussion to quasistatic hysteresis loops, i.e. nearly at local equilibrium. Dynamicand temperature effets require a specific discussion and microscopic modeling, whichwill be discussed in chapter sec.III, p.65.

Figure I.3 shows a typical hysteresis loop. We will speak of magnetization forthe sake of simplicity. However the concepts discussed more generally apply to anyother quantity involved in a hysteresis loop.

• Symmetry – Hysteresis loops are centro-symmetric, which reflect the time-reversal symmetry of Maxwell’s equations (H → −H and M → −M)

We will see in chapter V that hysteresis loops of certain heterostructuredsystems may be non-centro-symmetric, due to shifts along both the field andmagnetization axes. This however does not contradict the principle of time-reversal symmetry, as such hysteresis loops are minor loops. Application of asufficiently high field (let aside the practical availability of such a high field)would yield a centro-symmetric loop.

• ’Saturation’ magnetization – Due to Zeeman energy the magnetizationtends to align along the applied field when the magnitude of the latter islarge, associated with a saturation of the M(H) curve. For this reason oneoften names saturation magnetization the resulting value of magnetization. Wemay normalize the loop with its value towards saturation, and get a functionspanning in [−1; 1].


Two remarks shall be made. First the ’s’ subscript brings some confusionbetween spontaneous magnetization, which is a microscopic quantity used inmodels such as magnetic ordering and that one may like to determine exper-imentally, and the experimental quantity estimated at saturation of the loop.The knowledge of the volume of the system (if a moment is measured) or amodel (in case an experiment probes indirectly magnetization) is needed tolink an experimental quantity with a magnetization. Intrinsic or extrinsic con-tributions to the absence of true saturation of hysteresis loops are also an issue.

• Remanent magnetization – starting from the application of an externalmagnetic field, we call remanent magnetization (namely, which remains) andwrite Mr or mr when normalized, the value of magnetization remaining whenthe field is removed. After applying a positive (resp. negative) field, mr isusually found in the range [0; 1].

A negative remanence may occur in very special cases of heterostructuredsystems, as we will see in Chapter V.

• Coercive field – We call coercive field (namely, which opposes an action,here that of an applied magnetic field) and write Hc, the magnitude of fieldfor which the loop crosses the x axis, i.e. when the average magnetizationprojected along the direction of the field vanishes.

• Hysteresis and metastability – We have mentioned that the sign of rema-nence depends on that of the magnetic field applied previously. This featureis named hysteresis: the M(H) path followed for rising field is different fromthe descending path. Hysteresis results from the physical notion of metasta-bility: for a given magnitude (and direction) of magnetic field, there may existseveral equilibrium states of the system. These states are often only localminima of energy, and then said to be metastable. Coercivity and remanenceare two signatures of hysteresis. The number of degrees of freedom increaseswith the size of a system, and so may do the number of metastable states inthe energy landscape. The field history describes the sequence of magneticfields (magnitude, sign and/or direction) applied before an observation. Thishistory is crucial to determine in which stable or metastable state the sys-tem is leftI.5. This highlight the important role played by spatially-revolvedtechniques (both microscopies and in reciprocal space) to deeply characterizethe magnetic state of a system. Metastability implies features displayed dur-ing first-order transitions such as relaxation (over time) based on domain-wallmovement, nucleation and the importance of extrinsic features in these such asdefects. This implies that the modeling and engineering of the microstructureof materials is a key to control properties such as coercivity and remanence.

I.5The reverse is not true: it is not always possible to design a path in magnetic field liable toprepare the system in an arbitrary metastable state


• Energy losses – We often read the name magnetic energy, for a quantityincluding the Zeeman energy. This is improper from a thermodynamic pointof view. The Zeeman quantity −µ0M.H is the counterpart of +PV for fluidsthermodynamics: H is the vectorial intensive counterpart of pressure, and Mis the vectorial extensiveI.6 counterpart of volume, i.e. a response of the systemto the external stimulus. Thus, we should use the name density of magneticenthalpy for the quantity Eint − µ0M.H, where Eint is the density of internalmagnetic energy of the systemI.7, with analogy to H = U+PV . A readily-seenconsequence is that the quantity +µ0H.dM, analogous to−PdV , is the densityof work provided by the (external) operator and transferred to the system uponan infinitesimal magnetization process. Rotating the magnetization loop by90 to consider M as x, we see that the area encompassed by the hysteresisloop measures the amount of work provided to the system upon the loop, oftenin the form of heat (Figure I.3b).

• Functionalities of magnetic materials – The quantities defined above al-low us to consider various types of magnetic materials, and their use for appli-cations. Metastability and remanence are key properties for memory applica-tions such as hard disk drives (HDDs), as its sign keeps track of the previouslyapplied field, defining so-called up and down states. Coercivity is crucial forpermanent magnets, which must remain magnetized in a well-defined directionof the body with a large remanence, giving rise to forces and torques of crucialuse in motors and actuators. In practice coercivities of one or two Teslas maybe reached in the best permanent-magnet materials such as SmCo5, Sm2Co17

and Nd2Fe14B. The minimization of losses in the operation of permanent mag-nets and magnetic memories is important, both to minimize heating and forenergy efficiency. Among applications requiring small losses are transformersand magnetic shielding. To achieve this one seeks both low coercivity and lowremanence, which defines so-called soft magnetic materials. These materialsare also of use in magnetic field sensors based on their magnetic suscepti-bility, providing linearity (low hysteresis) and sensitivity (large susceptibilitydM/dH). A coercivity well below 10−3 A/m (or 1.25 mT in terms of µ0H)is obtained in the best soft magnetic materials, typically based on Permalloy(Fe20Ni80). On the reverse, some applications are based on losses such as in-duction stoves. There the magnitude of coercivity is a compromise betweenachieving large losses and the ability of the stove to produce large enough acmagnetic fields to reverse magnetization. Finally, in almost all applicationsthe magnitude of magnetization determines the strength of the sought effect,such as force or energy of a permanent magnet, readability for sensors andmemories, energy for transformers and induction heating.

• Partial loops – In order to gain more information about the magnetic mate-rial than with a simple hysteresis loop, one may measure a first magnetizationloop (performed on a virgin or demagnetized sample) or a minor loop (alsocalled partial loop or recoil loop), see Figure I.3a.

I.6or more precisely, the magnetic moment of the entire system∫VMsdr.

I.7see part 3 for the description of contributions to Eint

I.2. Units in Magnetism 15

We call intrinsic those properties of a material depending only on its composition andstructure, and extrinsic those properties related to microscopic phenomena related toe.g. microstructure (crystallographic grains and grain boundaries), sample shape etc.For example, spontaneous magnetization is an intrinsic quantity, while remanence andcoercivity are extrinsic quantities.

1.4 Domains and domain walls

Hysteresis loop, described in the previous section, concern a scalar and integratedquantity. It may thus hide details of magnetization (a vector quantity) at the mi-croscopic level. Hysteresis loops must be seen as one out of many signature of mag-netization reversal, not a full characterization. Various processes may determine thefeatures of hysteresis loops described above. It is a major task of micromagnetismand magnetic microscopies to unravel these microscopic processes, with a view toimprove or design new materials.

For instance remanence smaller than one may result from the rotation of mag-netization or from the formation of magnetic domains etc. Magnetic domains arelarge regions where in each the magnetization is largely uniform, while this directionmay vary from one domain to another. The existence of magnetic domains was pos-tulated by Pierre Weiss in his mean field theory of magnetism in 1907, to explainwhy materials known to be magnetic may display no net moment at the macroscopicscale. The first direct proof of the existence of magnetic domains came only in 1931.This is due to the bitter technique, where nanoparticles are attracted by the loci ofdomain walls. In 1932 Bloch proposes an analytical description of the variation ofmagnetization between two domains. This area of transition is called a magneticdomain. The basis for the energetic study of magnetic domains was proposed in1935 by Landau and Lifshitz.

Let us discuss what may drive the occurrence of magnetic domains, whereasdomain walls imply a cost in exchange and other energies, see sec.5. There existstwo reasons for this occurrence, which in practice often take place simultaneously.The first reason is energetics, where the cost of creating domain walls is balanced bythe decrease the dipolar energy which would be that of a body remaining uniformlymagnetized. This will be largely developed in chap.II. The second reason is magnetichistory, which we have already mentioned when discussing hysteresis loops (seesec.1.3). For instance upon a partial demagnetization process up to the coercivefield, domain walls may have been created, whose propagation will be frozen uponremoval of the magnetic field.

2 Units in Magnetism

The use of various systems of units is a source of annoyance and errors in magnetism.A good reference about units is that by F. Cardarelli[8]. Conversion tables formagnetic units may also be found in many reference books in magnetism, such asthose of S. Blundell[1] and J. M. D. Coey[3]. We shall here shortly consider threeaspects:

• The units – A system of units consists in choosing a reference set of elemen-


tary physical quantities, allowing one to measure each physical quantity witha figure relative to the reference unit. All physical quantities may then be ex-pressed as a combination of elementary quantities; the dimension of a quantitydescribes this combination. For a long time many different units were used,depending on location and their field of use. Besides the multiples were notthe same in all systems. The wish to standardize physical units arose duringthe French revolution, and the Academy of Sciences was in charge of it. In1791 the meter was the first unit defined, at the time as the ten millionth ofthe distance between the equator and a pole. Strictly speaking four types ofdimensions are enough to describe all physical variables. A common choice is:length L, mass M, time T, and electrical current I. This lead to the emergenceof the MKSA set of units, standing for Meter, Kilogram, Second, Ampere forthe four above-mentioned quantities. The Conferences Generale des Poids etMesures (General Conference on Weighs and Measures), an international or-ganization, decided of the creation of the Systeme International d’Unites (SI).In SI, other quantities have been progressively appended, which may in princi-ple be defined based on MKSA, however whose independent naming is useful.The three extra SI units are thermodynamic temperature T (in Kelvin, K),luminous intensity (in candela, cd) and amount of matter (in mole, mol). Thefirst two are linked with energy, while the latter is dimensionless. Finally,plane angle (in radian, rad) and solid angles (in steradian, sr) are called sup-plementary units. Another system than MKSA, of predominant use in thepast, is the cgs system, standing for Centimeter, Gramm, and Second. Atfirst sight this system has no explicit units for electrical current or charge,which is a weakness with respect to MKSA, e.g. when it comes to check thedimension homogeneity of formulas. Several sub-systems were introduced toconsider electric charges or magnetic moments, such as the esu (electrostaticunits), emu (electromagnetic units), or the tentatively unifying Gauss sys-tem. In practice, when converting units between MKSA and cgs in magnetismone needs to consider the cgs-Gauss unit for electrical current, the Biot (Bi),equivalent to 10 A. Other names in use for the Biot are the abampere or theemu ampere. Based on the decomposition of any physical quantity in ele-mentary dimensions, it is straightforward to convert quantities from one toanother system. For magnetic induction B 1 T is the same as 104 G (Gauss),for magnetic moment µ 1 A.m2 is equivalent to 103 emu and for magnetizationM 1 A/m is equivalent to 10−3 emu/cm3. In cgs-Gauss the unit for energy iserg, equivalent to 10−7 J. The issue of units would remain trivial, if restrictedto converting numerical valus. The real pain is that different definitions existto relate H, M and B, as detailed below.

• Defining magnetic field H – In SI induction is most often defined with B =µ0(H+M), whereas in cgs-Gauss it is defined with B = H+4πM . The dimen-sion of µ0 comes out to be L.M.T−2.I−2, thus µ0 = 4π×10−7 m.kg.s−2.A−2 in SI.Using the simple numerical conversion of units one finds: µ0 = 4π cm.g.s−2.Bi−2.Similar to the absence of explicit unit for electrical current, it is often arguedthat µ0 does not exist in cgs. The conversion of units reveals that one mayconsider it in the definition of M , with a numerical value 4π. However the def-inition de H differs, as the same quantity is written µ0H in SI, and (µ0/4π)Hin cgs-Gauss. Thus, the conversion of magnetic field H gives rise to an extra

I.3. The various types of magnetic energy 17

4π coefficient, beyond powers of ten. This pitfall explain the need to use anextra unit, the Oersted, to express values for magnetic field H in cgs-Gauss.Then 1 Oe is equivalent to (103/4π) A/mI.8. A painful consequence of thedifferent definitions of H is that susceptibility χ = dM/dH differs by 4π be-tween both systems, although is is a dimensionless quantity: χcgs = (1/4π)χSI.The same is true for demagnetizing coefficients defined by Hd = −NM , withNcgs = 4πNSI.

• Defining magnetization M – we often find the writing J = µ0M in theliterature. More problematic is the (rather rare) definition to use Ms insteadof µ0Ms. It is for instance the case of the book of Stohr and Siegmann[9],otherwise a very comprehensive book. These authors use the SI units, howeverdefine: B = µ0H + M. This can be viewed as a compromise between cgs andSI, however has an impact on all formulas making use of M .

This section highlights that, beyond the mere conversion of numerical values, formulasdepend on the definition used to link magnetization, magnetic field and induction. Itis crucial to carefully check the system of units and definition used by authors beforecopy-pasting any formulas implying M , H or B.

3 The various types of magnetic energy

3.1 Introduction

There exists several sources of energy in magnetic systems, which we review in thissection. For the sake of simplicity of vocabulary we restrict the following discussionto ferromagnetic materials, although all aspects may be extended to other types oforders. These energies will be described in the context of micromagnetism.

Micromagnetism is the name given to the investigation of the competition be-tween these various energies, giving rise to characteristic magnetic length scales, andbeing the source of complexity of distributions of magnetization, which will be dealtwith in chap.II.

Micromagnetism, be it numerical or analytical, is in most cases based on twoassumptions: :

• The variation of the direction of magnetic moment from (atomic) site to siteis sufficiently slow so that the discrete nature of matter may be ignored. Mag-netization M and all other quantities are described in the approximation ofcontinuous medium: they are continuous functions of the space variable r.

• The norm Ms of the magnetization vector is constant and uniform in anyhomogeneous material. This norm may be that at zero or finite temperature.The latter case may be viewed as a mean-field approach.

Based on these two approximations for magnetization we often consider the func-tion unitary vector function m(r) to describe magnetization distributions, such thatMs(r) = Msm(r).

I.8In practice, the absence of µ0 in the cgs system often results in the use of either Oersted orGauss to evaluate magnetic field and induction.


3.2 Zeeman energy

The Zeeman energy pertains to the energy of magnetic moments in an externalmagnetic field. Its density is:

EZ = −µ0M.H (I.7)

EZ tends to favor the alignement of magnetization along the applied field. Asoutlined above, this term should not be considered as a contribution to the internalenergy of a system, however as giving rise to a magnetic enthalpy.

3.3 Magnetic anisotropy energy

The theory of magnetic ordering predicts the occurrence of spontaneous magneti-zation Ms, however with no restriction on its direction in space. In a real systemthe internal energy depends on the direction of Ms with the underlying crystallinedirection of the solid. This arises from the combined effect of crystal-field effects(coupling electron orbitals with the lattice) and spin-orbit effects (coupling orbitalwith spin moments).

This internal energy is called magnetocrystalline anisotropy energy, whose den-sity will be written Emc in these notes. The consequence of Emc is the tendency formagnetization to align itself along certain axes (or in certain planes) of a solid, calledeasy directions. On the reverse, directions with a maximum of energy are called hardaxes (or planes). Magnetic anisotropy is at the origin of coercivity, although thequantitative link between the two notions is complex, and will be introduced inchap.II.

The most general case may be described by a function Emc = Kf(θ, ϕ), where fis a dimensionless function. In principle any set of angular functions complying withthe symmetry of the crystal lattice considered may be used as a basis to expressf and thus Emc. Whereas the orbital functions Yl,m of use in atomic physics maybe suitable, in practice one uses simple trigonometric functions. Odd terms do notarise in magnetocrystalline anisotropy because of time-reversal symmetry. Grouptheory can be used to highlight the terms arising depending on the symmetry of thelattice.

For a cubic material one finds:

Emc,cub = K1cs+K2cp+K3cp2 + . . . (I.8)

with s = α21α

22 + α2

2α23 + α2

3α21 and p = α2

1α22α

23. For hexagonal symmetry

Emc,hex = K1 sin2 θ +K2 sin4 θ + . . . (I.9)

where θ is the (polar) angle between M and the c axis. Here we dropped theazimuthal dependence because it is of sixth order, and that in practice the magnitudeof anisotropy constants decreases sharply with its order. Thus for an hexagonalmaterial the magnetocrystalline anisotropy is essentially uniaxial.

Group theory predicts the form of these formulas, however not the numericalvalues, which are material dependent. For example for Fe K1c = 48 kJ/m3 so thatthe < 001 > directions (resp. < 111 >) are easy (resp. hard) axes of magnetization,while for Ni K1c = −5 kJ/m3 so that < 001 > (resp. < 111 >) are hard (resp. easy)

I.3. The various types of magnetic energy 19

axes of magnetization. In Co K1 = 410 kJ/m3 and the c axis of the hexagon is thesole easy axis of magnetization.

In many cases one often considers solely a second-order uniaxial energy:

Emc = Ku sin2 θ (I.10)

It is indeed the leading term around the easy axis direction in all above-mentionedcases. We will see in sec.4 that it is also a form arising in the case of magnetostaticenergy. It is therefore of particular relevance. Notice that it is the most simpletrigonometric function compatible with time-reversal symmetry and giving rise totwo energy minima, this liable to give rise to hysteresis. It is therefore sufficient forgrasping the main physics yet with simple formulas in modeling.

Materials with low magnetic anisotropy energy are called soft magnetic materials, whilematerials with large magnetic anisotropy energy are called hard magnetic materials. Thehistorical ground for these names dates back to the beginning of the twentieth centurywhere steel was the main source of magnetic material. Mechanically softer materialswere noticed to have a coercivity lower than that of mechanically harder materials.

One should also consider magnetoelastic anisotropy energy, written Emel. Emel is themagnetic energy associated with strain (deformation) of a material, either compressive,extensive or shear. Emel may be viewed as the derivative of Emc with respect to strain.In micromagnetism the anisotropy energy is described phenomenologically, ignoring allmicroscopic details. Thus we may consider the sum of Emc and Emel, written for instanceEa or EK , a standing for anisotropy and K for an anisotropy constant.

3.4 Exchange energy

Exchange energy between neighboring sites may be written as:

E12 = −JS1.S2 (I.11)

i i+1a

θ

Figure I.4: Expansion of exchangewith θ to link discrete exchange tocontinuous theory.

J is positive for ferromagnetism, andtends to favor uniform magnetization. Let usoutline the link with continous theory usedin micromagnetism. We consider the text-book case of a (one-dimensional ) chain ofXY classical spins, i.e. whose direction ofmagnetization may be described by a singleangle θi (Figure I.4). The hypothesis of slowvariation of θi from site to site legitimates theexpansion:

E12 = −JS2 cos(δθ)

= −JS2

[1− (δθ)2

2

]= Cte +

JS2a2

2

(dθ

dx

)2

(I.12)


This equation may be generalized to a three dimensional system and moments al-lowed to point in any direction in space. Upon normalization with a3 to express adensity of energy, and forgetting about numerical factors related to the symmetryand number of nearest neighbors, one reaches:

Eex = A (∇m)2 . (I.13)

We remind the reader that m(r) is the unit vector field describing the magneti-zation distribution. The writting (∇m)2 is a shortcut for

∑i

∑j(∂xjmi)

2, linked

to Eq. (I.12). A is called the exchange, such as A ≈ (JS2/2a). It is then clear thatthe unit for A is J/m, which we find also in Eq. (I.13). The order of magnitude ofA for common magnetic materials such as Fe, Co and Ni is 10−11 J/m.

3.5 Magnetostatic energy

Magnetostatic energy, also called dipolar energy and written Ed, is the mutualZeeman-type energy arising between all moments of a magnetic body through theirstray field (itself called dipolar field and written Hd). When considering as a systeman infinitesimal moment δµMsδV the Zeeman energy provides the definition forenthalpy. However when considering the entire magnetic body as both the sourceof all magnetic field (dipolar field Hd) and that of moments, this term contributesto the internal energy.

Dipolar energy is the most difficult contribution to handle in micromagnetism.Indeed, due to its non-local character is may be expressed analytically in only avery restricted number of simple situations. Its numerical evaluation is also verycostly in computation time as all moments interact with all other moments; thiscontributes much to the practical limits of numerical simulation. Finally, due tothe non-uniformity in direction and magnitude of the magnetic field created by amagnetic dipole, magnetostatic energy is a major source of the occurrence of non-uniform magnetization configurations in bulk as well as nanostructured materials,especially magnetic domains. For all these reasons we dwell a bit on this term inthe following section.

3.6 Characteristic quantities

In the previous paragraphs we introduced the various sources of magnetic energy,and discussed the resulting tendencies on magnetization configurations one by one.When several energies are at play balances must be found and the physics is morecomplex. This is the realm of micromagnetism, the investigation of the arrangementof the magnetization vector field and magnetization dynamics. It is a major branchof nanomagnetism, and will be largely covered in chap.II.

It is a general situation in physics that when two or more effects compete, char-acteristic quantities emerge such as energy or length scales, and ratios. Here thesewill be built upon combination of Ms and H, a K constant such as Ku and A,which have different units. Characteristic length scales are of special importance innanomagnetism, determining the size below which specific phenomena occur. Herewe only make two preliminary remarks; more will discovered and discussed in thenext chapter, ending with an overview.

I.4. Handling dipolar interactions 21

Let us assume that in a problem only magnetic exchange and anisotropy compete.A and Ku are expressed respectively in J/m and J/m3. The only way to combinethese quantities to express a length scale, which we expect to arise in the problem,is ∆u =

√A/Ku. We will call ∆u the anisotropy exchange length[10] or Bloch

parameter as often found in the literature. This is a direct measure of the widthof a domain wall where magnetization rotates (limites by exchange) between twodomains whose direction is set by Ku.

In a problem where exchange and dipolar energy compete, the two quantities atplay are A and Kd = (1/2)µ0M

2s . In that case we may expect the occurrence of

the length scale ∆d =√A/Kd =

√2A/µ0M2

s , which we will call dipolar exchangelength[6] or exchange length as more often found in the literature.

In usual magnetic materials ∆u ranges from roughly one nanometer in the caseof hard magnetic materials (high anisotropy), to several hundreds of nanometers inthe case of soft magnetic materials (low anisotropy). ∆d is of the order of 10 nm.

4 Handling dipolar interactions

4.1 Simple views on dipolar interactions

To grasp the general consequences of Hd let us first consider the interaction betweentwo pinpoint magnetic dipoles µ1 and µ2, split by vector r. Their mutual energyreads (see sec.I.5):

Ed = − µ0

4πr3

[3

(µ1.r)(µ2.r)

r2− µ1.µ2

](I.14)

We assume both moments to have a given direction z, however with no constrainton their sign, either positive or negative. Let us determine their preferred respectiveorientation, either parallel or antiparallel depending on their locii, that of µ2 beingdetermined by vector r and the polar angle θ with respect to z (Figure I.5). EquationI.14 then reads:

E12 =µ0µ1µ2

4πr3(1− 3 cos2 θ) (I.15)

The ground state configuration being the one minimizing the energy, we see thatparallel alignement is favored if cos2 θ > 1/3, that is within a cone of half-angle θ =54.74, while antiparallel alignement is favored for intermediate angles (Figure I.5).Thus, under the effect of dipolar interactions two moments roughly placed alongtheir easy axis tend to align parallel, while they tend to align antiparallel whenplaced next to each other. These rules rely on angles and not the length scale, andare thus identical at the macroscopic and microscopic scales. The example is thatof permanent magnets, which are correctly approached by Ising spins.

The occurrence of a large part of space where antiparallel alignement is favored(outside the cone) makes us feel why bulk samples may be split in large blockswith different (e.g. antiparallel) directions of magnetization. These are magneticdomains. Beyond these handwavy arguments, the quantitative consideration ofdipolar energy is outlined below in the framework of a continuous medium.


4.2 Ways to handle dipolar fields

z

rθ

Figure I.5: Interaction be-tween two Ising spins ori-ented along z. Parallel(resp. antiparallel) aligne-ment is favored inside (resp.outside) a cone of half-angle54.74.

The total magnetostatic energy of a system withmagnetization distribution M(r) reads :

Ed = −µ0

2

∫M(r).Hd(r) d3r. (I.16)

The pre-factor 12

results from the need not to counttwice the mutual energy of each set of two elemen-tary dipoles taken together. The decomposition ofa macroscopic body in elementary magnetic mo-ments and performing a three-dimensional integralis not a practical solution to evaluate Ed. It is oftenbetter to proceed similarly to electrostatics, withdiv E = ρ/ε0 being replaced by div Hd = −div M(derived from the definition of B, and Maxwell’sequation div B = 0). Within this analogy, ρ =−div M are called magnetic volume charges. A lit-tle algebra shows that the singularity of div M thatmay arise at the border of magnetized bodies (Ms

going abruptly from a finite value to zero on eitherside of the surface of the body) can be lifted by introducing the concept of surfacecharges σ = M.n. One has finally:

Hd(r) =

∫ρ(r′) (r− r′)

4π|r− r′|3d3r′ +

∮σ(r′) (r− r′)

4π|r− r′|3.dS (I.17)

dS with S oriented towards the outside of the body is the elementary integrationsurface. This set, notice that Hd has a zero rotational and thus derives from apotential: Hd = −gradφd. Equation I.16 may then worked out, integrating inparts:

Ed =1

2µ0

∫M(r).gradφd(r) d3r (I.18)

=1

2µ0

∫Mi(r).∂xiφd d3r (I.19)

=

[1

2µ0φdMi

]∞− 1

2µ0

∫(∂xiMi)φd d3r (I.20)

(I.21)

The first term cancels for a finite size system, and one finds a very practical formu-lation:

Ed =1

2µ0

(∫ρφd d3r +

∫σφd dS

). (I.22)

Another equivalent formulation may be demonstrated:

Ed =1

2µ0

∫H2

d d3r (I.23)


where integration if performed other the entire space. From the latter we infer thatEd is always positive or zero. Equation I.22 shows that if dipolar energy alone siconsidered, its effect is to promote configurations of magnetization free of volumeand surface magnetic charges. Such configurations are thus ground states (possiblydegenerate) in the case where dipolar energy alone is involved.

• The tendency to cancel surface magnetic charges implies a very general rule forsoft magnetic materials: their magnetization tends to remain parallel to the edgesand surfaces of the system.

• The name dipolar field is a synonym for magnetostatic field. It refers to all mag-netic fields created by a distribution of magnetization or magnetic moments inspace. The name stray field refers to that part of dipolar field, occurring outsidethe body responsible for this field. The name demagnetizing field refers to thatpart of dipolar field, occurring inside the body source of this field; the explanationfor this name will be given later on.

The term dipolar brings some confusion between two notions. The first notionis dipolar (field or energy) in the general sense of magnetostatic. The name dipolarstems from the fact that to compute total magnetostatic quantities of a magnetic body,whatever its complexity, one way is to decompose it into elementary magnetic dipoles andperform an integration; the resulting calculated quantities are then exact. The secondnotion is magnetic fields or energies arising from idealized pinpoint magnetic dipoles, andobeying Eq. (I.14). When using the name dipolar to refer to the interactions betweentwo bodies, one may think either that we compute the exact magnetostatic energy basedon the integration of elementary dipoles, or that we replace the two finite-size bodieswith pinpoint dipoles for the sake of simplicity, yielding on the reverse an approachevaluation. In that latter case one may add extra terms, called multipolar, to improvethe accuracy of the approximation. To avoid confusion one should stress explicitlythe approximation in the latter case, for instance mentioning the use of apoint dipole approximation.

4.3 Demagnetizing factors

Demagnetizing factors (or coefficients) are a simple concept providing figures for themagnetostatic energy of a body. Eq. (I.17) applied to uniform magnetization retainsonly the surface contribution

Hd(r) = Ms

∫(r− r′)

4π|r− r′|3mini dS(r′) (I.24)

with M ≡Msm, m = miui and n = niui, with Einstein’s summation notation. n isthe local normal to the surface, oriented towards the outside of the body. Injectingthis equation into Eq. (I.16) yields after straightforward algebra a compact formulafor the density of demagnetizing energy:

Ed = Kdtm.N.m (I.25)

with Kd = 12µ0M

2s , and N a 3× 3 matrix with coefficients:


Table I.3: Demagnetizing factors for cases of practical use

Case Demagnetizing factor Note

Slab Nx = −1 Normal along x

General ellipsoid Nx = 12abc

∫∞0

[(a2 + η)

√(a2 + η)(b2 + η)(c2 + η)

]−1dη

Prolate revolution ellipsoid Nx = α2

1−α2

[1√

1−α2arg sinh

(√1−α2

α

)− 1

]α = c/a < 1

Oblate revolution ellipsoid Nx = α2

α2−1

[1 − 1√

α2−1arcsin

(√α2−1

α

)]α = c/a > 1

Cylinder with elliptical section Nx = 0, Ny = c/(b + c) and Nz = b/(b + c) Axis along x

Prism Analytical however long formula See: [6] or [13]

Nij =

∫d3r

∫ni(u) (r− r′)j

4π|r− u|3dS(u) (I.26)

N is called the demagnetizing matrix. It may be shown that N is symmetric andpositive, and thus can be diagonalized. The set of xyz axes upon diagonalization arecalled the main or major axes. The coefficients N ′ii of the diagonal matrix are calledthe demagnetizing coefficients and will be written Ni hereafter as a shortcut. Alongthese axes it is readily seen that the following is true for the average demagnetizingfield, providing a simple interpretation of demagnetizing factors:

〈Hd,i〉 = −NiM. (I.27)

N yield a quadratic form, so that only second-order anisotropies can arise fromdipolar energy, at least for perfectly uniform samplesI.9.

It can be shown that Tr(N) = 1, so that Nx + Ny + Nz = 1. Analytical formu-las for Ni’s may be found for revolution ellipsoids[11], prisms[12, 13] (Figure I.6),cylinders of finite length[14, 15, 16], and tetrahedrons[17, 18]. Some formulas aregathered in Table I.3. For other geometries micromagnetic codes or Fourier-spacecomputations[18] may be used.

While all the above is true for bodies with an arbitrary shape, not even necessarilyconnected, a special subset of bodies is worth considering: that of shapes embodied by apolynomial surface of degree at most two. To these belong slabs, ellipsoids and cylinderswith an ellipsoidal cross-section. In that very special case it may be shown within thenon-trivial theory of integration in space[19] that Eq. (I.27) is then true locally: in thecase of uniform magnetization, Hd is uniform and equal to −NiM when M is alignedparallel to one of the major directions. This allows the torque on magnetization to bezero, and thus ensures the self-consistency of the assumption of uniform magnetization.This makes the application of demagnetizing factors of somewhat higher reliability thanfor bodies with an arbitrary shape. Notice, however, that self-consistency does notnecessarily imply that the uniform state is stable and a ground state.

I.9see sec.4.4 for effects due to non-uniformities


0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

p=b/a

1/20

1/5

1/2

1/125 (cylinder)

Nz

Nx

x=c/a

xa

c

b

y

z

p=

p=

p=1/20

p=1/20

x=c/a

0.4

0.5

0.6

0.7

0.8

0.9

1

0.02 0.04 0.06 0.08 0.1

0.02

0.04

0.06

0.08

0.1

0.12

(a) (b)

Figure I.6: Numerical evaluation of demagnetizing factors for prisms. (a) is the fullplot, while (b) is en enlargement for flat prisms.


Demagnetizing factors are derived based on the assumption of uniform magnetization.While this assumption allows demagnetizing factors to be defined and calculated analyt-ically or numerically, care should be taken when applying these to practical cases, wheremagnetization configurations may not be uniform.

5 The Bloch domain wall

The existence of magnetic domains was suggested by Pierre Weiss in his meanfield theory of Magnetism in 1907. Magnetic domains were postulated to explainwhy large bodies made of a ferromagnetic materials could display no net magneticmoment under zero external magnetic field. Their existence was confirmed onlyin 1931 with a bitter technique, based on magnetic nanoparticles decorating thelocii of domain walls because these particles are attracted by the local gradient ofmagnetic field. This example highlights the importance of magnetic microscopy inthe progress of micromagnetism. In 1932 Bloch provides an analytical solution ina simple case to describe the region of transition between two magnetic domains,which is named a magnetic domain wall. At this stage we do not discuss the originof magnetic domains, however focus on the model of a domain wall.

The Bloch model is one-dimensional, i.e. considers a chain of spins. The ideais to describe the transition between two three-dimensional domains (volumes) inthe form of a two-dimensional object with translational invariance in the plane ofthe domain wall. It is assumed that magnetization remains in the plane of thedomain wall, a configuration associated with zero volume charges −div M and thusassociated zero dipolar energy. The only energies at play are then the exchangeenergy, and the magnetic anisotropy energy which is assumed to be uniaxial and ofsecond order. Under these assumptions the density of magnetic energy reads:

E(x) = Ku sin2 θ + A (∂xθ)2 (I.28)

where x if the position along the chain of spins. The case thus consists in exhibitingthe magnetic configuration which minimizes the total energy

E =

∫ +∞

−∞[EK(x) + Eech(x)]dx. (I.29)

while fulfilling boundary conditions compatible for a 180 domain wall: θ(−∞) = 0and θ(+∞) = π.

5.1 Simple variational model

This paragraph proposes an approached solution for a domain wall, however appeal-ing for its simplicity and ability to highlight the physics at play, and a reasonablenumerical result. We consider the following model for a domain wall of width `:θ = 0 for x < −`/2, θ = π(x + `/2) for x ∈ [−`/2; `/2] and θ = π for x > `/2. Ina variational approach we search for the value `var which minimizes Eq. (I.29), afterintegration: E = Ku`/2 + Aπ2/`. The minimization yields `var = π

√2√A/Ku and

Evar = π√

2√AKu is the associated energy.

I.5. The Bloch domain wall 27

Letting aside the factor π√

2 a simple variational model highlights the relevance of theBloch parameter ∆u defined previously. How may we read this formula? Exchange onlywould tend to enlarge the domain wall, hence its occurrence at the numerator. To thereverse, the anisotropy energy gives rise to a cost of energy in the core of the domainwall. This tends to decrease its width, explaining its occurrence at the denominator.

5.2 Exact model

The exact profile of a Bloch domain wall may be derived using the principle offunctional minimization to find the function θ minimizing E . It may be shown thatthe principle of minimization is equivalent to the so-called Euler equation:

∂E

∂θ=

d

dx

[∂E

∂( dθdx

)

](I.30)

Using a condensed notation this reads:

∂θE = dx(∂(dxθ)E

)(I.31)

Considering a magnetic system described by Eq. (I.29) one finds:

dθEK = dx (2A∂xθ) (I.32)

= 2A∂xxθ (I.33)

Upon multiplying both parts by ∂xθ and integration, this reads:

EK(x)− EK(a) = A [∂xθ(x)]2 − A [∂xθ(a)]2

= Eex(x)− Eex(a) (I.34)

a is the origin of integration, here chosen as the center of the domain wall. Consid-ering two semi-infinite domains with equal local density of energy, E is stationary(minimum) in both domains, and by convention may be chosen zero with no loss ofgenerality. Equation I.34 applied to ±∞ shows that EK(a) = Eex(a), and finally:

∀x EK(x) = Eex(x) (I.35)

We hereby reach a general and very important feature of a domain wall separatingtwo semi-infinite domains under zero applied field: the local density of anisotropyand exchange energy are equally parted at any location of the system. Theequal parting of energy considerably eases the integration to get the areal densityof the domain wallI.10:

I.10We set arbitrarily ∂xθ > 0 without loss of generality, using the symmetry x→ −x.


Magnetization angle

-8 -6 -4 -2 0 2 4 6 80

π

π/2

Distance (in Δ units)

Figure I.7: Exact solution for the profile of the Bloch domain wall (red dots), alongwith its asymptote (red line). The lowest-energy solution of the linear variationalmodel is displayed as a black line.

E = 2

∫ +∞

−∞A (dxθ)

2 dx

= 2

∫ +∞

−∞EK(x) dx

= 2

∫ +∞

−∞

√AEK(x)dxθ dx

= 2

∫ θ(+∞)

θ(−∞)

√AEK(θ) dθ (I.36)

The energy of the domain wall may thus be expressed from the anisotropy of en-ergy alone, without requiring solving the profile of the domain wall, which may beinteresting to avoid calculations or when the latter cannot be solved.

Let us come back to the textbook case of the functional I.28. After some algebraone finds for the exact solution:

θex(x) = 2 arctan [exp(x/∆u)] (I.37)

Eex = 4√AKu. (I.38)

∆u =√A/Ku is of course confirmed to be a natural measure for the width of

a domain wall. The exact solution along with that of the variational model aredisplayed on Figure I.7. Despite its crudeness, the latter is rather good, for both thewall profile and its energy: the true factor afore

√AKu equals 4 against π

√2 ≈ 4.44

in the variational model. It is trivial to notice that Evar > Eex, as the energy ofa test function may only be larger than the energy of the minimum functional. Itshall be noticed that the equal parting of energy is retained in the variational model,however only in its global form, not locally.

5.3 Defining the width of a domain wall

Several definitions for the width δW of a domain wall have been proposed (see e.g.Ref.[6], p.219).

I.6. Magnetometry and magnetic imaging 29

The most common definition was introduced by Lilley[20]. It is based on theintercept of the asymptotes (the domain) with the tangent at the origin (the wall)of the curve θ(x). This yields δL = π

√A/Ku = π∆u for the exact solution, and

δL = `variationel =√

2∆u for the linear variational model.

Some call ∆u the domain wall width. To avoid any confusion is it advised to keep thename Bloch parameter for this quantity.

A second definition consists in using the asymptotes of the curve cos θ(x), insteadof that of θ(x). On then finds δm = 2

√A/Ku, both in the exact and variational

models.A third definition is δF =

∫ +∞−∞ sin θ(x)dx. In the present case of a uniaxial

anisotropy of second order one finds δF = δL.The latter two definitions are more suited for the analysis of domain walls in-

vestigated by magnetic microscopies probing the projection of magnetization in agiven direction. Besides, δF is based on an integration. It can thus be applied toany type of domain wall, whereas the definitions of Lilley and δm may be ambiguousin materials with high anisotropy constants, with domain wall profiles potentiallydisplaying several inflexion points.

The use of cos and sin fonctions in the definitions δm and δF is dependent on the startingand ending angles of the domain wall, here 0 and π. For other choices or domain wallswith angle differing from 180, these definitions shall be modified.

6 Magnetometry and magnetic imaging

There exists many techniques to probe magnetic materials. Due to the smallamounts to be probed, and the need to understand magnetization configurations,high sensitivity and/or microscopies are of particular interest for nanomagnetism.There exists no such thing as a universal characterization technique, that would besuperior to all others. Each of them has its advantages and disadvantages in terms ofversatility, space and time resolution, chemical sensitivity etc. The combination ofseveral such techniques is often beneficial to gain the full understanding of a system.

Here a quick and non-exhaustive look is proposed over some techniques that haveproven useful in nanomagnetism. In-depth reviews may be found elsewhere[6, 21,22, 23].


Sample

FB ON

FB OFF

Figure I.8: Principle of the two-pass procedure usually implemented for MFM imag-ing.

6.1 Extraction magnetometers

6.2 Faraday and Kerr effects

6.3 X-ray Magnetic Dichroism techniques

6.3.a X-ray Magnetic Circular Dichroism

6.3.b XMCD Photo-Emission Electron Microscopy

6.3.c XMCD Transmission X-ray Microscopy

6.4 Near-field microscopies

6.4.a Magnetic Force Microscopy

Magnetic Force Microscopy (MFM) is derived from Atomic Force Microscopy, forwhich good reviews are available. Along with Kerr microscopy, it is the most pop-ular magnetic microscopy technique owing to its combination of moderate cost,reasonable spatial resolution (routinely 25-50 nm) and versatility. Good reviews areavailable for both AFM[24] and MFM[22, 21].

AFM and MFM probe forces between a sample and a sharp tip. The tip is non-magnetic in the former case, and coated with a few tens of nanometers of magneticmaterial in the latter case. The forces are estimated through the displacement of asoft cantilever holding the tip, usually monitoring the deflection of a laser reflectedat the backside of the cantilever. The most common working scheme of MFM isan ac technique: while the cantilever is mechanically excited close to its resonancefrequency f0 (or more conveniently written as the angular velocity ω0 = 2πf0), thephase undergoes a shift proportional to the vertical gradient of the (vertical) force∂F/∂z felt by the tip: ∆ϕ = −(Q/k)∂zF . In practice magnetic images are gath-ered using a so-called two-pass technique: each line of a scan is first conducted inthe tapping mode with strong hard-sphere repulsive forces probing mostly topogra-phy (so-called first pass), then a second pass is conducted flying at constant height(called the lift height) above the sample based on the information gathered duringthe first pass. Forces such as Van der Waals are assumed to be constant during thesecond pass, and the forces measured are then ascribed to long-range forces such asmagnetic.

I.6. Magnetometry and magnetic imaging 31

The difficult point with MFM is the interpretation of the images, and the pos-sible mutual interaction between tip and sample. A basic discussion of MFM isproposed in the Problems section, p.37. A summary of the expected signal mea-sured is provided in Table I.4.

Table I.4: Expected MFM signal with respect to the vertical component Hd,z of thestray field in static (cantilever deflection) and dynamic (frequency shift during thesecond pass) modes versus the model for the MFM tip.

Tip model Static response Dynamic responseMonopole Hd,z ∂Hd,z/∂zDipole ∂Hd,z/∂z ∂2Hd,z/∂z

2

6.4.b Spin-polarized Scanning Tunneling Microscopy

6.5 Electron microscopies

6.5.a Lorentz microscopy

6.5.b Scanning Electron Microscopy with Polarization Analysis (SEMPA)

6.5.c Spin-Polarized Low-Energy Electron Microscopy (SPLEEM)

Problems for Chapter I

Problem 1: More about units

Here we derive the dimensions for physical quantities of use in magnetism, and theirconversions between cgs-Gauss and SI.

1.1. Notations

We use the following notations:

• X is a physical quantity, such as force in F = mg. It may be written X forvectors.

• [X] is the dimension of X. As a shortcut we will use a vector to summarize thepowers of the fundamental units length (L), mass (M), time (T) and electricalcurrent (I). For example, speed and electrical charges read: [v] = [L] − [T ] =[1 0 −1 0] and [q] = [I] + [T ] = [0 0 1 1]. We use shortcuts [L], [M ], [T ] and[I] for the four fundamental dimensions.

• In a system of units α (e.g. SI or cgs-Gauss) a physical quantity is evaluatednumerically based on the unit physical quantities: X = Xα〈X〉α. Xα is a num-ber, while 〈X〉α is the standard (i.e., used as unity) for the physical quantityin the system considered. For example 〈L〉SI is a length of one meter, while〈L〉cgs is a length of one centimeter: 〈L〉SI = 100〈L〉cgs. For derived dimensionswe use the matrix notation. For example the unit quantity for speed in systemα would be written [1 0 − 1 0]α.

1.2. Expressing dimensions

• Based on laws for mechanics, find dimensions for force F , energy E andpower P , and their volume density E and P .

• Based on the above, find dimensions for electric field E, voltage U , resis-tance R, resistivity ρ, permittivity ε0.

• Find dimensions for magnetic field and magnetization H and M, induction Band permeability µ0.

32

Problem 2: More about the Bloch domain wall 33

1.3. Conversions

Physics does not depend on the choice for a system of units, so doesn’t anyphysical quantity X. The conversions between its numerical values Xα and Xβ

in two such systems is readily obtained from the relationship between 〈X〉α and〈X〉β, writing: X = Xα〈X〉α = Xβ〈X〉β. Let us consider length l as a example.l = lSI〈L〉SI = lcgs〈L〉cgs. As 〈L〉SI = 100〈L〉cgs we readily have: lSI = (1/100)lcgs.Thus the numerical value for the length of an olympic swimming pool is 5000 incgs, and 50 in SI. For derived units (combination of elementary units), 〈X〉α isdecomposed in elementary units in both systems, whose relationship is known. Forexample for speed: 〈v〉α = 〈L〉α〈T 〉−1

α . Notice that in the cgs-Gauss system, theunit for electric charge current may be considered as existing and named Biot orabampere, equivalent to 10 A.

Exhibit the conversion factor for these various quantities, of use for magnetism:• Energy E , energy per unit area Es, energy per unit volume E. The unit for

energy in the cgs-Gauss system is called erg.

• Express the conversion for magnetic induction B and magnetization M , whoseunits in cgs-Gauss are called Gauss and emu, respectively. Express relatedquantities such as magnetic flux φ and magnetic moment µ.

• Let us recall that magnetic field is defined in SI with B = µ0(H+M), whereasin cgs-Gauss with B = H + 4πM , with the unit called Oersted. Express theconversion for µ0 and comment. Then express the conversion for magneticfield H.

• Discuss the cases of magnetic susceptibility χ = dM/dH and demagnetizingcoefficients defined by Hd = −NM .

Problem 2: More about the Bloch domain wall

The purpose of this problem is to go deeper in the mathematics describing thetextbook case of the Bloch domain wall discussed in sec.5. We recall the followingshortcuts: ∂xθ for ∂θ/∂x and ∂nxθ for ∂nθ/∂xn.

2.1. Euler-Lagrange equation

We will seek to exhibit a magnetization configuration that minimizes an energydensity integrated over an entire system: E =

∫E(r)dr. Finding the minimum of

a continuous quantity integrated over space is a common problem solved throughEuler-Lagrange equation, which we will deal with in a textbook one-dimensionalframework here.

Let us consider a microscopic variable defined as F (θ, dxθ), where x is the spatialcoordinate and θ a quantity defined at each point. In the case of micromagnetismwe will have:

F (θ, dxθ) = A (dxθ)2 + E(θ) (I.39)

34 Problems for Chapter I

When applied to micromagnetism E(θ) may contain anisotropy, dipolar andZeeman terms. We define the integrated quantity:

F =

∫ B

A

F (θ, dxθ) dx+ EA(θ) + EB(θ). (I.40)

A and B are the boundaries of the system, while EA(θ) and EB(θ) are surfaceenergy terms.

Let us consider an infinitesimal function variation δθ(x) of θ. Show that extremaof F are determined by the following relationships:

∂θF − dx (∂dxθF ) = 0 (I.41)

dθEA − ∂dxθF |A = 0 (I.42)

dθEB + ∂dxθF |B = 0 (I.43)

Notice that equations Eq. (I.42) and Eq. (I.43) differ in sign because a surfacequantity should be defined with respect to the unit vector normal to the surface,with a unique convention for the sense, such as the outwards normal. Here theabscissa x is outwards for pointB however inwards at A. An alternative microscopicexplanation would be that for a given sign of dxθ the exchange torque exerted on amoment to the right (at point B) is opposite to that exerted to the left (at pointA), whereas the torque exerted by a surface anisotropy energy solely depends on θ.

2.2. Micromagnetic Euler equation

Apply the above equations to the case of micromagnetism [Eq. (I.39)]. Startingfrom Eq. (I.41) exhibit a differential equation linking E[θ(x)] with dxθ. EquationsI.42-I.43 are called Brown equations. EA(θ) and EB(θ) may be surface magneticanisotropy, for instance. Discuss the microscopic meaning of these equations.

Comment the special case of free boundary conditions (all bulk and surfaceenergy terms vanish at A and B), in terms of energy partition. Now on we switchback to the physics notation E for the total energy, instead of F . Show that it canbe expressed as:

E = 2

∫ θ(B)

θ(A)

√AE(θ) dθ (I.44)

2.3. The Bloch domain wall

Let us assume the following free boundary conditions, mimicking two extendeddomains with opposite magnetization vectors separated by a domain wall whoseprofile we propose to derive here: θ(−∞) = 0 and θ(+∞) = π. We will assume thesimplest form of magnetic anisotropy, uniaxial of second order: E(θ) = Ku sin2 θ.

Based on a dimensional analysis give approximate expressions for both the do-main wall width δ and the domain wall energy E . What are the SI units for E?Discuss the form of these quantities in relation with the meaning and effects ofexchange and anisotropy.

Problem 2: More about the Bloch domain wall 35

Magnetization angle

-8 -6 -4 -2 0 2 4 6 80

π

π/2

Distance (in Δ units)

Figure I.9: Bloch domain wall profile: the exact solution (red dots) versus theasymptotic profile (red line). The solution with linear ersatz is shown as a dark line.

By integrating the equations exhibited in the previous section, derive now theexact profile of the domain wall:

θ(x) = 2 arctan[exp(x/∆)] (I.45)

and its total energy E .

The most common way to define the Bloch domain wall width δBl is by replacingthe exact θ(x) by its linear asymptotes (red line on Figure I.9). To shorten theexpressions we often use the notation ∆ =

√A/Ku, called the Bloch parameter.

Derive δBl as a function of ∆.

Let us stress two issues:• The model of the Bloch wall was named after D. Bloch who published this

model in 1932[25].

• As often in physics we have seen in this simple example that a dimensionalanalysis yields a good insight into a micromagnetic situation. It is alwaysworthwhile starting with such an analysis before undertaking complex ana-lytical or numerical approaches, which especially for the latter may hide thephysics at play.

• We have exhibited here a characteristic length scale in magnetism. Otherlength scales may occur, depending on the energy terms in balance. Thephysics at play will often depend on the dimensions of your system with respectto the length scales relevant in your case. Starting with such an analysis isalso wise.

• when the system has a finite size the anisotropy and exchange energy do notcancel at the boundaries. The integration of Euler’s equations is more tedious,involving elliptical functions.


Problem 3: Extraction and vibration magnetome-

ter

3.1. Preamble

Here we consider the principle of extraction magnetometry, either in full quasi-dcextraction operation, or in the vibration mode (Vibrating Sample Magnetometer,VSM). Their purpose is to estimate the magnetic moment held by a sample, possiblyas a function of field, temperature, time etc. The general principle is to move asample along the axis of a coil of radius R. This induces a change over time ofthe flux in the coil, arising from the sample, which may be measured thanks tothe induced electromotive force (EMF)I.11. In a so-called extraction magnetometerthe sample is moved sufficiently away from end to the other along the axis so asto nearly cancel the flux, resulting in an absolute measurement of the flux. In avibrating sample magnetometer the sample vibrates along the axis at several tensof hertz close to the coil, inducing a large EMF and opening the use of a lock-in technic to further reducing the noise, however the full extraction curve is notmeasured, resulting in higher sensitivity to artefacts, as will be discussed below.

3.2. Flux in a single coil

Based on the Biot and Savart formula, express as a vector the induction B(z)arising along the axis of a circular coil of radius R with electrical current I. Belowis reminded the Biot and Savart formula expressing at an arbitrary location M inspace the infinitesimal induction δB arising from a current I on an infinitesimalelement δl at location P :

δB =µ0Iδl(P )×PM

4πPM3(I.46)

For reaching a high sensitivity the coil is wound several time, N 1. In thefollowing we will assume N = 1000 for numerics. We will assume here that thelocation of all loops is the same. Based on the reciprocity theorem for induction,derive the magnetic flux Φ(z) in the series of coils, arising from a pinpoint magneticmoment µ located on the axis of the coil. Φ(z) will be expressed as Φ(z) = Kf(z),with f(z) a dimensionless function. Draw a schematics of f(z)Numerics: what is the uniform magnetic induction that would be required tocreate a flux in these coils, equivalent to that of a square piece of thin film ofiron of lateral size 1 cm and thickness 1 nm (reminder: the magnetization of iron is≈ 1.73×106 A/m). Comment with respect to the magnitude of the earth magneticfield.

3.3. Vibrating in a single coil

The sample is now moved periodically along the axis of the coil, around thelocation z0: z(t) = z0 + ∆z cos(ωt). Based on a first-order expansion in ∆z/z0,

I.11An alternative and very sensitive device for measuring the flux through a coil is SQUID:Superconducting Quantum Interference Device

Problem 4: Magnetic force microscopy 37

derive the EMF e(t) induced in the coil. Draw a schematics of this curve. At whichposition is found the maximum of magnitude for e(t)?Numerics: calculate the magnitude of e(t) arising from the iron thin film mentionedabove with a frequency of 30 Hz and ∆z = 1 mm. Comment about this value.

3.4. Noise in the signal

Figure I.10: Geometry fortwo coils winded in oppo-site directions

Owing to a mechanical coupling the coils for mea-surement vibrate with angular frequency ω in the sup-posedly static induction B applied to magnetize thesample. Let us assume that due the coils’ imperfec-tions or finite size this induction displays an inhomo-geneity ∆B at the spatial scale for vibration of thesample. Derive the EMF induced in the measuringcoils due to this inhomogeneity.Numerics: vibration of magnitude 1µm in an induc-tion of strength 1 T, with a relative change of 10−3

over a distance of 5 mm. Comment the value.

3.5. Winding in opposition

The above noise can be reduced by using two coilswith same axis, measured in series however wound inopposite senses (Figure I.10). The measured EMF isthen etot(t) = e2(t)−e1(t), and the sample is vibratedat equal distance from the two coils, at the position z0

such that the signal is maximum (see above). Why isthe above noise significantly reduced? Comment thissetup with respect to the Helmoltz geometry for twocoils.

Problem 4: Magnetic force microscopy

This problem is an extension of the short paragraph about magnetic force mi-croscopy in this chapter. This paragraph should be read first, before addressing thisproblem.

4.1. The mechanical oscillator

The dynamics of the AM cantilever is modeled by a mechanical oscillator:

md2z

dt2+ Γ

dz

dt+ k(z − z0) = F (z, t) (I.47)

F (z, t) is a force arising from either the operator or from the tip-sample in-teraction, and z0 is the equilibrium position without applied force. m, Γ and kare the oscillator mass, damping and stiffness, respectively. We use the notationω0 =

√k/m and Q =

√km/Γ, the latter being called the quality factor.

Rewrite Eq. (I.47) with the use of ω0 and Q. The cantilever is excited by theoperator with F (t) = Fexce

jωt. Provide the transfer function H = z/F , the gain G =|H| and phase shift ϕ = arg(H), as well as the following quantities, at resonance:angular velocity ωr, magnitude zr and phase ϕr. For the case Q 1 calculate the


magnitude at resonance, and the full-width at half maximum (FWHM) ∆ωr of theresonance peak. Comment.

4.2. AFM in the static and dynamic modes

The cantilever is brought in the vicinity of the surface, inducing a non-zero forceF (z) between the tip and sample, adding up to the sinusoidal from the operator.For the sake of simplicity we will model the variations of F using a simple affinefunction: F (z) = F (z0) + (z − z0)∂zF .

Calculate the new position at equilibrium zeq. Rewrite Eq. (I.47) in this case, andin the case Q 1 the normalized change of resonance angular velocity δωr/ω0. Inmost cases the cantilever is excited at a constant frequency ωexc and the force gradi-ent is monitored through the change of frequency ∆ϕ. Show that ∆ϕ = (Q/k)∂zF .

4.3. Modeling forces

We assume here that the magnetization configurations of both the tip and thesample are not influenced one by another. The vertical component of the forceapplied by the sample on the tip is F = −∂zE , where E is the mutual energy. Thetip may be modeled either by a magnetic dipole µ, or by a magnetic monopole q) (inpractice tips may be modeled by a linear combination of both components). Forboth models express to which z derivative of the vertical component of the samplestray field Hd,z are proportional the deflection in the static AFM mode, and thefrequency shift in the dynamic AFM mode.

Numerical evaluation – A typical MFM cantilever has Q = 1000 and k =4 N/m. Modeling both the tip and samples by a magnetic dipole made of Co with adiameter 25 nm, and assuming a probing distance of 50 nm, provide a crude estimateof the frequency shift expected. Comment.

Chapter II

Magnetism and magnetic domainsin low dimensions

1 Magnetic ordering in low dimensions

1.1 Ordering temperature

The main feature of a ferromagnetic body is spontaneous ordering below a criticaltemperature TC, called Curie temperature. It was Weiss who first proposed a mean-field approach to describe the ordering. In this theory it is postulated that the localmoments feel an internal magnetic field

Hi = nWMs + H (II.1)

where H is the external field, and nWMs is the co-called molecular field. This is aphenomenological representation of magnetic exchange, whose quantum-mechanicalorigin was not known at the time. A semi-classical description allows to link theHeisenberg hamiltonian H = −2

∑i>j Ji,jSi.Sj with nW:

2ZJi,j = µ0nWng2JµB

2 (II.2)

where Z is the number of nearest neighbors, n the volume density of sites, eachholding a dimensionless spin S bounded between −J and +J , associated with totalmagnetic moment µJ = gJJµB

II.1. Based on the site susceptibility related to theBrillouin functionBJ , the expected ordering temperature may be expressed as:

TC =2ZJi,jJ(J + 1)

3kB

(II.3)

The expected Curie temperature is therefore proportional to Z. Let us now drawtrends for the Curie temperature in low dimensions. To do this we consider athin film as a model system, and extend the mean-field approach to averaging thenumber of nearest neighbors over the entire system. For a film with N layers of siteswith magnetic moments we get: ZN = Z + 2(Zs − Z)/N where Zs is the numberof nearest neighbors of each of the two surface/ interface layers (Figure II.1a). AsZs < Z we immediately see based on Eq. (II.3) that the ordering temperature should

II.1Beware of this local possible confusion between the exchange constant J , and the total angularmomentum J .

39

40 Chapter II. Magnetism and magnetic domains in low dimensions

Figure II.1: Magnetic ordering in low dimensions, here with N = 5 atomic layers.(a) Counting the reduced average number of nearest neighbors in a thin film withN atomic layers. Example of an experimental determination of (b) the temperaturedependence of magnetization and (c) the Curie temperature in various ultrathin filmmaterials[27].

be reduced with a 1/t law. Our handwavy considerations are confirmed by a morerigorous layer-dependent mean-field theory[26]. Going beyond mean-field, one mayfind other critical exponents λ for TC ∼ t−λ. .

As a rule of thumb, following Eq. (II.3) TC should be decreased to half thebulk ordering temperature for N equaling one or two atomic layers. Figure II.1b-cshows the Ms(T ) variation and the Curie temperature measured for several typesof ultrathin films, where the latter prediction appears largely valid, although thescaling law is best fitted with λ = 1.27± 0.20.

Finally, the Ms(T ) law again depends on the model used (dimensionality, typeof moment, ordering model), and so do critical exponents in both limits of T → 0+

and T → TC−. In the low temperature range the decay is dominated by spin waves

and follows a Bloch law:

Ms(T ) = Ms(T = 0 Ku)[1− bNT 3/2] (II.4)

whereas mean-field theory predicts an exponentially-weak decay. bN is the spin-wave parameters, which again happens to be thickness-dependent and well fittedwith a 1/t law[27]. The case of a truly two-dimensional system should clearly betreated on a different footing due to the absence of out-of-plane excitations. WhileOnsager derived an expression for the finite Curie temperature in a 2d array ofIsing spins[28], the Mermin and Wagner theorem states that long-range ordering isnot expected to occur at finite temperature for a 2D array of Heisenberg spins; thedivergence of susceptibility is found only for T → 0 K. This problem has long excitedexperimentalists, with no report of absence of ferromagnetism in a 2d system. Thereason is that an energy gap is opened in the spin-wave spectrum as soon as magneticanisotropy sets in, of magnetocrystalline origin[29] or even simply magnetostatic[30].

Said in a handwavy fashion, any source of anisotropy mimics Ising spins at sufficientlylow temperature, going in the direction of the Onsager solution.

II.1. Magnetic ordering in low dimensions 41

Figure II.2: Schematics of the effect of band narrowing on Stoner criterium and themagnitude of the magnetic moment.

In one dimension thermal fluctuations have an even stronger impact, leading toabsence of ordering at any finite temperature even for Ising spins. Thus the corre-lation length is not expected to diverge until truly zero temperature. Experimentalresults pertaining to such systems is available and indeed points at the existence offinite-size spin blocks[31].

1.2 Ground-state magnetic moment

Here we discuss the magnitude of the ground-state spontaneous magnetization atzero temperature. The case of itinerant magnetism in 3d metals is particularly welldocumented, and the general trend is physically interesting. Let us consider the caseof a free-standing layer, i.e. with no supporting nor capping material. Due to theloss of coordination at both surfaces, 3d bands are expected to narrow (Figure II.2).As the total number of electrons is conserved this should help satisfying Stonercriterium Iρ(εF) where I is the exchange integral and ρ(εF) the density of electronsfor each spin channel. This in turn should enhance the imbalance of the number ofoccupied states in both spin channels, and thus magnetization. This trend may beunderstood as moving towards free electron magnetism were Hund’s rules apply andorbital momentum is not quenched, hence giving rise to a larger magnetic momentper atom. In most systems this trend is confirmed through ab initio calculationsand observed experimentally[27]. Exceptions (reduction of moment with respect tothe bulk) may be explained by phenomena whose consequences are more difficultto predict such as epitaxial or surface strain, dislocations, hybridization and chargetransfer with an interfacial material, quantum-size effects. . . . Mainly the latter playa role in more localized magnetism, leading to effects more difficult to predict.

Thin films are easy to model and simulate thanks to translational invariance.However low-dimensional effects arise equally in other systems such as clusters. Themagnetic moment per atom has been measured to be clearly enhanced in these, evi-denced in-flight with Stern-Gerlach experiments or capped with sensitive techniquessuch as XMCD[32]. The Stoner criterium may even be fulfilled in clusters, while itis not in the bulk form. A famous case is Rhodium[33, 34].


Conclusion

We have reviewed the basics of ferromagnetic ordering in low dimensionsfor itinerant magnetism. The general trend is that of two competingeffects. The zero-temperature ground state displays a moment gener-ally larger than that of the bulk, due to band narrowing. An oppositetrend if the enhanced decay of magnetization with temperature. At fi-nite temperature both effects compete, requiring care in the analysis ofmeasurements.

2 Magnetic anisotropy in low dimensions

We first consider magnetostatic anisotropy, long-ranged and related to the outershape of a system. We then consider the magnetic anisotropy of microscopic origin,arising from spin-orbit and the crystal electric field. These are magnetocrystallineand magnetoelastic anisotropies, which were introduced in sec.3. We consider thinfilms as a model system, however those concepts apply to all low-dimensional sys-tems, however in a more complex manner.

2.1 Dipolar anisotropy

In sec.4.3 we introduced the concept of demagnetizing factors. These were calculatedon the assumption that the system under consideration is uniformly magnetized.Although this may be questionable in some cases even under applied field, in thepresent section we will rely on these factors for a first discussion. In this frameworkwe have seen [Eq. (I.25)] that the dipolar contribution to magnetic anisotropy reads,after proper diagonalization defining the so-called main directions of anisotropy:Ed = −KdNimi, and the internal so-called demagnetizing f477ield reads Hd =−NiMiui, where i runs over all three main directions, and Nx +Ny +Nz = 1.

For thin films Nx = Ny = 0 along the two in-plane directions, resulting inzero demagnetizing field and demagnetizing energy. Nz=1, resulting in Ed = Kd

and Hd = −Msuz for perpendicular magnetization. The resulting demagnetizinginduction µ0Ms is of the order of one Tesla for common materials (Table I.2).

Unless the material displays a very large microscopic energy, or a very strong field isapplied perpendicular to the plane, the magnetization of a thin film lies preferentiallyin-the-plane.

For cases other than films, however of reduced dimension in at least one direction,we will speak of nanostructures. The demagnetizing factors are all three non-zero,and again if no microscopic energy or applied field applies, the magnetization willhave a tendency to point along the direction with the lowest demagnetizing factors.

Let us add a fine point often subject to controversy, however of great importancefor domains and magnetization reversal in nanostructures: the range of dipolar in-teractions. Dipolar interactions are commonly described as long-ranged. This isso because the stray field from a magnetic dipolar decays with distance like 1/r3.Thus, an upper bound for the stray field at a given location is of type

∫(1/r3)4πr2dr,

summing over the entire system magnitudes instead of vectors. This diverges log-arithmically (however converges if vectors are considered instead of magnitudes),

II.2. Magnetic anisotropy in low dimensions 43

Figure II.3: Definition of axes for a cubic crystal projected along the (110) plane.

revealing the long range of dipolar fields. More precisely, it is straightforward toshow that what matters is the solid angle under which a surface density of chargesis seen, not its distance. Let us now consider a flat system, for instance an el-ement patterned out of a thin film with lithography. The upper bound becomes∫

(1/r3)2πrdr, which converges to a finite value with a radius of convergence scalingwith the sample thickness. In other words:

Dipolar energy is short-ranged in two dimensions. This can be understood in a handwavymanner as most stray field escapes in the third dimension, not contributing to the selfenergy −(1/2)µ0MsHd. This implies that stray- and demagnetizing fields are oftenhighly non-homogeneous, with important consequences on both magnetization patternsand magnetization reversal processes. For the same reason, the concept of demagnetizingfactors and shall be used with great care in such cases

Elements with two flat surfaces (made out of a thin film) and with a circular or ellipsoidshape are not ellipsoids. Their demagnetizing field is therefore highly non-uniform, asfor all flat elements.

2.2 Projection of magnetocrystalline anisotropy due to dipo-lar energy

One consequence of magnetostatic energy is to favor the alignement of magnetizationin directions with small demagnetizing coefficients. If magnetostatic energy prevailsover magnetocrystalline anisotropy energy, the magnetization will tend to lie incertain planes or directions imposed by the former, while the latter will play a roleonly through its projection in these planes or directions. Let us consider the exampleof a cubic material; its magnetocrystalline anisotropy is described by Eq. (I.8), whosemagnitude is measured through the parameter K1c. If K1c is much smaller than Kd

then the direction of magnetization will be imposed by the latter, for instance in-the-plane for a thin film (sec.4.3). As an example, let us consider a cubic crystalcut along a (110) plane (Figure II.3). When restricted to ϕ = 90, Eq. (I.8) reads:

Emc,cub = K1c sin2 ϕ+ (−3

4K1c +

1

4K2c +K3c) sin4 ϕ+ . . . (II.5)

Then, the effective anisotropy in the plane becomes uniaxial.


We illustrated a feature of symmetries with application to many fields in physics, suchas bulk versus surface crystallography: considering of a function defined in a space withd dimensions and displaying certains symmetries, its projection or restriction into asub-space of dimension lower than d does not necessarily preserve or restrict the initialsymmetry, even if the sub-space is an element of symmetry of the initial function.

2.3 Interface magnetic anisotropy

The local environment of atoms differs at both surfaces of a thin film with respectto the bulk one. In 1954 L. Neel suggested that this breaking of symmetry inducedby the loss of translational invariance along the normal to the film, should resultin an additional term to magnetic anisotropy. This was well before technologyenabled to produce films so thin and well characterized that experiments couldsuggest the effect. This additional term is called surface magnetic anisotropy, orinterface magnetic anisotropy, or also Neel magnetic anisotropy II.2.

As for magnetocrystalline anisotropy, interface anisotropy may favor an easy di-rection or an easy plane, and be decomposed in angular terms with various orders.As it applies only once per each interface, its effects becomes vanishingly small atlarge thickness. In practice it is observed that its effect becomes negligible beyonda few nanometers. One speaks of ultra thin films in this range smaller than char-acteristic length scales, where magnetization is obviously allowed to vary along thethickness. At a given lateral position its magnetizationII.3 may be described as a sin-gle vector, the so-called macrospin, on which apply both surface and bulk magneticanisotropy. As a simple example let us assume that both terms are uniaxial alongthe same axis, with two identical surfaces. The resulting anisotropy then readsKvt + 2Ks with Kv and Ks the volume and surface contributions. The effectivedensity of energy thus reads:

Keff = Kv +2Ks

t(II.6)

Following this, the usual way to estimate Ks in theory and experiments is to plot Keff

versus 1/t. The intercept with the y axis should yield the bulk anisotropy, while theslope should yield Ks (Figure II.4). Interfacial anisotropies between various types ofmaterials has thus been tabulated[27, 36]. Ks indeed depends on the material, maybe of different sign, and is of the order of 0.1 mJ.m−2.

In its 1954 model Neel proposed the estimation of an order of magnitude forKs values, based on the phenomenological analogy between removing the atoms tocreate an interface, and pulling them away infinitesimally. Ks was then linked withmagneto-elastic constants of the material, with surprisingly a good agreement onthe order of magnitude, although the exact value and even the sign may be wrong.The so-called pair model of Neel aims at describing the direction and material-

II.2In principle interface is appropriate to describe a thin magnetic film in contact with anothermaterial while surface is appropriate to describe a free surface (in contact with vacuum). Thislatter case is in principle restricted to fundamental investigations performed in situ in UHV, wherea surface may remain free of contaminant for some time. In practice, both terms are often usedinterchangeablyII.3More precisely its moment per unit area, thus expressed in Amperes


dependence of surface anisotropy by counting the bounds between a surface atomand the neighbors, and associate them with a uniaxial angular function.

Figure II.4: A historical exampleof 1/t plot for evaluating interfa-cial anisotropy[35].

Theory can also be used to evaluate Ks val-ues. Letting aside ab initio calculations, for 3dmetals tight binding links magnetocrystallineanisotropy with the anisotropy or the orbitalmagnetic moment. For a uniaxial anisotropythe energy per magnetic atom is:

κ = αξ

4µB

∆µL. (II.7)

ξ is the spin-orbit coupling, defined by con-tribution −ξS.L to the Hamiltonian. ∆µL isthe difference of orbital magnetic moment be-tween hard and easy directions, and α is a fac-tor close to unity and only weakly related withthe details of the band structure.

In bulk 3d metals the orbital momentumis nearly fully quenched because crystal elec-tric field energy dominates over spin-orbit, andeigen functions in a cubic symmetry shouldhave nearly zero orbital momentum. Thus∆µL are very weak, typically of the order of10−4 µB/atom, yielding K ≈ 104 J/m3. Atboth surfaces and interfaces this anisotropy is enhanced close to 0.1µB/atome, in-ducing an anisotropy of energy of the order of 1 meV per surface atom, which liesclose to 1 mJ/m2. The link between surface magnetic anisotropy and ∆µL hasbeen checked experimentally and by ab initio calculations to be essentially valid.Some experiments hint at a quantitative link between bulk and surface magneticanisotropy[37], however the universality of this link remains speculative.

The most dramatic consequence of surface magnetic anisotropy, with also oftechnological use, arises when Ks favors the alignement of magnetization along thenormal to a thin film : Es = Ks cos2(θ) with Ks < 0 and θ the angle between mag-netization and the normal to the film. If Eq. (II.6) is negative and becomes greaterin absolute value than Kd for a realistic critical value of thickness tc, magnetizationwill point spontaneously along the normal to the film. This is perpendicular mag-netic anisotropy (PMA). For a long time the most efficient interfaces to promotePMA combined 3d elements for the ferromagnet, and a heavy element to bring inspin-orbit. Prototypical examples are Co/Au, Co/Pt and Co/Pd. tc is of the orderof 2 nm or less. Recently even larger contributions to perpendicular anisotropy, andthus larger critical thicknesses (up to 3.5 nm), have been reported at the interfacebetween 3d metals and oxides, with the prototypical case of Co/MgO. If films thickerthan this are needed with perpendicular magnetization, a route is the fabrication ofmultilayers[36].


2.4 Magnetoelastic anisotropy

The concept of magnetic surface anisotropy has been presented above as a textbookcase. In fact it is not the single source of modification of magnetic anisotropyin ultrathin films. We review here an equally important source, magnetoelasticanisotropy.

In the bulk form strain may be obtained through stress applied by an externaluser. Strain is always present in thin films to some extent even at rest. This is dueto the effect of the supporting material (and to some smaller extent the cappingmaterial), which having a lattice parameter and possibly symmetry different fromthat of the overgrown magnetic material, stresses the latter. Stress may also appearupon cooling (resp. warming up) thin films fabricated at high (resp. low) tempera-ture. This results in a strain field in the magnetic film, generally not uniform, whichgives rise to a magnetoelastic contribution to the total MAE.

One should not confuse strain with stress. The former is the deformation, the latter isthe force related to the strain.

To first order magnetoelastic anisotropy is proportional to the matrix elementsof strain. Group theory predicts the type of coupling terms[38], not their strength.In thin films there clearly exists an asymmetry between out-of-plane and in-planedirections: stress is applied in the latter, while along the former the film is free torelax. This results in a uniaxial magnetoelastic contribution.

Let us understand the qualitative effect of magnetoelasticity in thin films using asimple model. We consider the epitaxial growth of a film material (lattice parameteraf) on a substrate (lattice parameter as), the latter being assumed to be rigid. Thelattice misfit is defined as η = (af−as)/as. During growth the deposited material willtend to relax its strain ε = (a− af)/af through, e.g., the introduction of interfacialdislocations. We further assume that the linear energy cost per dislocation k doesnot depend on the density of dislocations, and that each dislocation allows thecoincidence of N + 1 atoms of the film with N substrate atoms (resp., the reverse),which corresponds to negative (resp. positive) η. Working in a continuum model,the density of mechanical energy of the system is :

Emec =1

2Cε2 +

k

taf

|η + ε− ε2| (II.8)

where t is the film thickness and C a elastic constant. The equilibrium value for ais found through minimization of this equation with the constraint |ε| < |η|:

• Below the critical thickness tc = k/(asC|η|) the introduction is dislocation isunfavorable, and a = as. The layer is said to be pseudomorph. As a rule ofthumb, tc ≈ 1 nm for η ≈ 2 − 5 %. This value is however dependent on thecrystal symmetry, growth temperature and technique of deposition.

• Above ef dislocations are created and reduced strain, following: |ε(e)| =k/(asCt).

What what have described so far is a structural model, proposed in 1967 byJesser[39]. In 1989 Chappert et Bruno applied this model to magneto-elasticity[40].


They considered linear magneto-elastic termsII.4. As a simple case, let us assume thatall deformations may be expressed in terms of ε, so that Emel = Bε with B a couplingconstant. Based on the structural model of Jesser we derive: Kmel = kB/(asCt).Beyond the pseudomorphic regime we therefore expect a dependence of Kmel with1/t, thus exactly like for a contribution of magnetic interface anisotropy. In mostcases magneto-elasticity and surface anisotropy are intermingled in thin films; itis almost impossible experimentally and conceptually to disentangle them. Never-theless, it remains common to designate as surface anisotropy the total effectivecontribution revealed as a 1/t variation of the density of magnetic anisotropy.

2.4.a Anisotropy resulting from the synthesis process

Following the above, it might be expected that beyond a few nanometers of thickness,the anisotropy of thin films is similar to that of bulk. While this is often the case,there are cases of persistence for large thickness of a magnetic anisotropy differentfrom the bulk one.

A first reason is the the Jesser model considers the minimum of energy. Inpractice this minimum may not be reached perfectly due to the energy barriersrequired to create dislocations, and it is often the case that thin films retains afraction of percent if strain. The exact value strongly depends on the couple ofmaterials, the orientation of the grains, the conditions and technic of deposition.

A second reason for the persistence of deviations from bulk anisotropy is the oftenfine microstructure induced by the growth method. The microstructure may take theform of grains separated by grains boundaries, incorporated of foreign atoms (likeAr during sputtering growth), an anisotropic orientation of atomic bounds etc. Thiseffect has dramatic consequences for materials with large magnetostriction such as3d-4f compounds, which can be tailored to display perpendicular anisotropy for fairlythick films. It is also possible to tailor a uniaxial anisotropy between two in-planedirections, through deposition under an applied field like for Permalloy (Ni80Fe20),or deposition with oblique incidence or on a trenched surface. Another eleganttechnique to tailor the anisotropy of thin films is irradiation with ion of mediumenergy. This irradiation may be done during growth or post-growth. When theirradiation energy is suitably chosen, the ions may either favor the mixing of atomsor their segregation, depending on the thermodynamics trend for alloying on phaseseparation. Irradiating thin films with perpendicular anisotropy, the former leadsto a decrease of anisotropy, while the latter leads to an increase. Irradiation may becombined with masks to deliver films with patterned anisotropy, however no changesin topography[42].

Conclusion

Contributions to magnetic anisotropy of energy in thin films includemagnetostatic, magnetocrystalline, interfacial and magnetoelastic ener-gies. For very thin films the latter two often dominate in the nanometerrange of thickness, opening the way to beating dipolar anisotropy todisplay perpendicular magnetization.

II.4It was recently shown that non-linear effects may be important in thin films[41]. This effecthad not been reported in bulk materials, where plastic deformation sets in well before strain valueslarge enough for non-linearities may be reached


Figure II.5: Schematics for (a) a Bloch domain wall and (b) a Neel domain wall.

3 Domains and domain walls in thin films

3.1 Bloch versus Neel domain walls

In sec.6.5.csec.?? we considered a textbook case of domain-wall: the Bloch domainwall, resulting from the competition of exchange energy against magnetocrystallineanisotropy. A translational invariance along both directions perpendicular to thedomain wall was assumed, so that the problem boiled down to a unidimensionalequation that can be solved.

Translational invariance makes sense in the bulk, where domain walls may ex-tend laterally on distances much longer than their width. This hypothesis becomesquestionable in thin films, where the core of a Bloch domain wall, displaying perpen-dicular magnetization, induces the appearance of magnetic charges at both surfacesof the thin film (Figure II.5a).

L. Neel was first in addressing this issue and providing a rule-of-thumb predic-tion for a cross-over in the nature of domain walls in thin films[43]. In a thin film ofthickness t he considered a domain wall of bulk width w ≈ ∆u, such as determinedfrom exchange and anisotropy energies. He took into account the finite size effectalong the normal to the film, modeling the domain wall as a cylinder of perpendicularmagnetization with an elliptical cross-section of axes w×t (Figure II.5). For a Blochdomain wall the resulting density of magnetostatic energy is Kd w/(w + t), basedon demagnetizing coefficients (Table I.3). When t < w it becomes more favorablefor magnetization in the core of the domain wall to turn in-the-plane, for which thedensity of magnetostatic energy is Kd t/(w + t) (Figure II.5b). This configurationwhere magnetization turns in-the-plane, i.e. perpendicular to the domain wall, iscalled a Neel wall.

In the above model the core of the domain wall was assumed to be rigid and uni-formly magnetized. Besides, its energy was calculated crudely, and is not suitable forsoft magnetic materials where magnetostatic energy dominates magnetic anisotropyso that no natural width of the domain wall exists. The phase diagram of Blochversus Neel wall can then be refined using micromagnetic simulations. These showin the case of soft magnetic material that Neel walls become stable for thicknessbelow 7∆d (already below 15− 20∆d for cross-tie walls, see next paragraph)e.g. for50 nm for Permalloy and 20 nm for Fe[44].

Micromagnetic simulations also revealed a phase diagram more complex thanmerely Bloch versus Neel walls (Figure II.6a). Going towards large thicknesses do-main walls undergo a breaking of symmetry with respect to a vertical plane; they arenamed asymmetric Neel wall and asymmetric Bloch wall, and were first proposedin 1969 through both micromagnetic simulation[45] and an ersatz model[46]. Let us

II.3. Domains and domain walls in thin films 49

(a) (b)

Figure II.6: (a) phase diagram of domain walls in thin films, calculated for practicalreasons in a stripe of finite width[44] (b) one of the first success of micromagneticsimulation, predicting the existence of the asymmetric Bloch domain wall[45].

examine the detail of the asymmetric Bloch wall, of higher practical interest (Fig-ure II.6b). Close to the surface the magnetization turns in-the-plane; this may beunderstood from the necessity to eliminate surface magnetic charges to decreasemagnetostatic energy, or in other words to achieve a flux-closure state. The surfaceprofile of magnetization is similar to that of a Neel wall, later motivating the nameof Neel cap do designate this area of flux-closure[47]. Notice that the center of theNeel cap is displaced from the vertical of the core of the Bloch wall, explaining thename asymmetric for this domain wall. This asymmetry arises so as so reduce nowvolume magnetic charges, balancing ∂xmx with ∂zmz terms in the divergence of M.Close to the transition from Bloch to Neel the cross-section of the asymmetric Blochwall looks similar to a vortex, so that the name vortex wall is sometimes used.

3.2 Domain wall angle

We define as wall angle θ, the angle between the direction of magnetization in twoneighboring domains. The properties of a domain walls as a function its angledepend on parameters such as film thickness t, anisotropy strength and symmetry.Here we restrict the discussion to rather soft magnetic materials in rather thin films,so that most of the energy of a domain wall is of magnetostatic origin.

The density of volume charges in an extended domain wall is −∂xMx, where x isthe coordinate along the in-plane axis perpendicular to the domain wall (Figure II.7).Generally a wall is induced to bisect the direction of magnetization of the twoneighboring domains, so that is bears no net magnetic charge and thus does notcontribute significantly to magnetostatic energy through a long-range 1/r decay ofstray field (Figure II.7). Following Neel, we model the core of the domain wall witha cylinder of elliptical cross-section, and estimate its energy through the suitabledemagnetizing coefficient.

We first consider a Neel wall. The total quantity of charge in each half of theelliptical cylinder scales with 1− cos(θ/2), which can be replaced with a reasonableaccuracy with θ2/8. As dipolar energy scales with the square of charges, and as-


Figure II.7: Wall angle and magnetostatic charges. (a) A wall that would notbisect the direction of magnetization in the neighboring domains would bear a netcharge (b) A wall bisecting the magnetization directions in neighboring domains isassociated with a dipolar line.

suming that the domain wall width does not depend significantly on the wall angle,we come to the conclusion that the energy of a Neel domain wall varies like θ4.

We now consider a Bloch wall. Volume charges can be avoided if mx is uniformand equal to cos(θ/2) from one domain to the other, through the domain wall. Thismeans that, apart from the case θ = 180, the core of such a wall has both in-planeand out-of-plane components, the latter equal to

√1− cos2(θ/2) = sin(θ/2). Thus

the magnetostatic energy a Bloch wall scales like sin(θ/2) ≈ θ2/4.

The energy of a domain wall depends on its angle θ. In this films the energy of a Neelwall varies like θ4, much faster than that of a Bloch wall, varying like θ2.

3.3 Composite domain walls

Dramatic consequences result from the convex variation of domain wall energy withangle outlined above. To set ideas, the cost per unit length of a 90 Neel wall is lessthan 10 % that of a 180 Neel wall. This means that a 180 Neel wall may be unstableand be replaced by walls of smaller angle, even is this implies an increase of thetotal length of domain wall. This is confirmed experimentally with the occurrenceof composite domain walls.

One type of composite domain wall is the so-called cross-tie (Figure II.8a-b). Itcan be checked that each wall fulfils is bisecting the neighboring domains. Cross-tiedomain walls occur only in soft magnetic material, because the extended domainwith different orientations shall not come at the expense of an anisotropy energy.Notice also that as the energy of a Bloch wall scales like θ2 whereas that of a Neelscales like θ4 (see previous paragraph), 180 Bloch walls are replaced with cross-tiewalls for a thickness larger than that predicted by the Neel model for the cross-overbetween Bloch and Neel.

Another type of composite wall is the zig-zag domain wall. Although domainwalls tend to bisect the direction of neighboring domains, it may happen due to thehistory of application of field and nucleation of reversed domains, that two domainsface each other and are each stabilized, e.g. by a uniaxial anisotropy or a gradientof external field with opposite signs. A 180 is unstable as the net magnetostaticcharge carried would be Ms, the largest possible value. In this case the domainwall breaks into short segments connected in a zig-zag line (Figure II.8c-d). Along


Figure II.8: Composite domain walls in thin films: (a-b) Schematics and MFM image(13 × 15µm)[48] of a cross-tie wall. On the schematics open and full dots standfor vortices and antivortices, respectively (c-d) schematics and Kerr image(350 ×450µm)[6] of a zig-zag wall.

the segments the walls have a tendency to turn 180 to be free of volume charges,implying some continuous rotation of magnetization in the dihedron formed by twoconsecutive segments. The angle of the zig-zag is determined by a complex balancebetween the reduction of magnetostatic energy due to the net charge, versus theincrease of energy through the wall length, and anisotropy and exchange energy inthe domains.

3.4 Vortices and antivortex

The inspection of Figure II.8a reveals the existence of loci where, from symmetryand continuity arguments, the direction of magnetization may be in no direction inthe plane. These were called Bloch lines, consisting of a cylinder of perpendicularmagnetization separating two Neel walls with opposite directions of in-plane mag-netization. The direction of perpendicular magnetization in a Bloch line is calledthe polarity, and summarized by the variable p = ±1. Bloch lines also occur insideBloch walls, separating parts of the wall core with opposite directions of perpen-dicular magnetization. Thus Bloch lines are the one-dimensional analogous domainwalls, separating two objects of dimensionality larger by one unit. In Bloch linesexchange and dipolar energy compete, yielding a diameter scaling with ∆d, of theorder of 10 nm in usual materials.

It is useful to introduce the concept of winding number defined like:


n =1

2π

∫∂Ω

∇θ.d` (II.9)

where ∂Ω is a path encircling the Bloch line, and θ is the angle between the in-planecomponent of magnetization and a reference in-plane direction. Applied to the cross-tie wall, this highlights alternating Bloch lines with n = 1 and n = −1 (resp. openand full dots on Figure II.8a). The former are also called vortex and the latter anti-vortex. Notice that through the transformation of a translation-invariant Neel wallwith no Bloch line into a cross-tie wall, the total winding number is thus conserved.This is a topological property, which will be further discussed in the framework ofnanostructures (see sec.4).

We also introduce the chirality number:

c = − k

2π.

∫∂Ω

∇m× d` (II.10)

where k is the normal to the plane defining its chirality. On Figure II.8a) vorticeshave c = +1.

Bloch lines are fully characterized by three numbers: polarity p, winding number n andchirality c. An antivortex has zero chirality, while vortices have c = ±1 depending onthe sense of rotation of magnetization, either clockwise or anticlockwise.

There exists also a zero-dimensional object, the Bloch point, separating two parts of aBloch line with opposite polarities. For topological (continuity) reasons, at the centerof the Bloch point the magnitude of magnetization vanishes, making it a very peculiarobject[49].

3.5 Films with an out-of-plane anisotropy

Here we consider thin films with a microscopic contribution to the magnetic anisotropyenergy, favoring the direction perpendicular to the plane. Most depends on the qual-ity factor Q = Ku/Kd and film thickness t. For Q < 1 uniform in-plane magnetiza-tion is a (meta)stable state however with large energy, while uniform out-of-planemagnetization is not a (meta)stable state. For Q > 1 the situation is reversed. Inall cases a balance between anisotropy energy and shape anisotropy needs to befound, the best compromise being through non-uniform states. The competition ofall four energy terms leads to a rich phase diagram, see Ref.6 for a comprehensivetheoretical and experimental review. A schematic classification with no applied fieldis presented below, and summarized in Table II.1.

In the case of large thickness (see table and below for numbers), in all cases thestate of lowest energy is one of alternating up-and-down domains, with a period2W (Figure II.9a). This pattern is called strong stripe domains. This situation wasfirst examined by Kittel[50], and later refined by several authors. The alternancecancels surface charges on the average, keeping magnetostatic energy at a low level.Magnetic anisotropy is also kept at a low level as most of magnetization lies alongan easy direction. The remaining costs in energy arise first from the vertical domainwalls (of Bloch type with in-plane magnetization to avoid volume charges), second


Table II.1: Summary of the magnetization state of films with an out-of-plane con-tribution to magnetic anisotropy. t and W are the film thickness and the optimumdomain width, respectively.

Q < 1 Q > 1

t > tc Weak to strong stripe domains withincreasing t. W ∼ t1/2 and thenW ∼ t2/3 upon branching

Strong stripe domains. W ∼ t1/2

and then W ∼ t2/3 upon branching.May be hindered by hysteresis.

tc Second order transition (no hys-teresis in the case of purely uniax-ial anisotropy) from uniform in-the-plane to weak stripes

The minimum value for W isreached.

t < tc Uniform in-plane magnetization Perpendicular domains with diverg-ing W , however quickly masked byhysteresis.

from flux-closure slabs close to the surface with a complex mixture of anisotropy,dipolar and wall energy. Minimization of this energy yields straightforwardly anoptimum value for W scaling like

√t, more precisely like

√t√AKu/Kd for Q &

1 (Figure II.9a) and like√t√A/Ku for Q . 1 (Figure II.9b). At quite large

thicknesses[6], typically hundreds of nanometers or micrometers, this law is modifieddue to branching of domains close to the surface (Figure II.9c). Branching decreasesthe energy of closure domains, while saving wall energy in the bulk of the film. Wethen have W ∼ t2/3.

For decreasing thickness we shall consider separately two cases. For Q > 1 thereexists a critical minimum domain width Wc ≈ 15

√AKu/Kd, which is reached for

tc ≈ Wc/2. Below this thickness flux-closure between neighboring domains becomeslargely ineffective due to the flat shape of the domains, thereby leading to a sharpincrease of W , with ultimately a divergence for t → 0 (Figure II.9d). For Q < 1the magnetization in the domains progressively turns in-the-plane, with a second-order transition towards a uniform in-plane magnetization around t = 2π∆u. Thispattern is called weak stripe domains due to the low angle modulation of directionof magnetization in neighboring domains. Close to the transition W ≈ t and thedeviations from uniformity are sinusoidal to first order.

In the above, notice that the state with lowest energy may not be reached for Q > 1, asthe uniform state perpendicular to the plane is (meta)stable. Thus strong stripe domainsmay not occur even at large thickness, for very coercive materials. Below tc the energygain resulting from the creation of domains is very weak, so that the divergence of W isoften hidden again behind coercive effects.

Conclusion

The features of domain walls are different in thin films, compared to


Figure II.9: Stripe domains. Sketches for (a) open domains (initial Kittel’s model,(b) perfect flux-closure domains and (c) domain branching. (d) predicted width ofdomain W with film thickness t, from ref.6. lc = 2

√AKu/Kd

the bulk. This is mostly related to the need to reduce dipolar energy,arising because of the loss of translational invariance along the normalto the film. The thickness of the film has a strong impact, and oftenapproximations are required to describe the physics analytically.

4 Domains and domain walls in nanostructures

In this section we examine the effect of reducing the lateral dimensions of nanostruc-tures, from large to small nanostructures. We consider first the domains, followedby special cases of domain walls.

4.1 Domains in nanostructures with in-plane magnetization

We consider a piece of a thin film of soft magnetic material, quite extended howeverof finite lateral dimensions. Under zero applied field these assumptions allow us todescribe the arrangement of magnetization as an in-plane vector field m of normunity, and neglect the energy inside and between domain walls. Under these condi-tions Van den Berg proposed a geometrical construction to exhibit a magnetizationdistribution with zero dipolar energy[51, 52]. As dipolar energy is necessary zero orpositive, this distribution is a ground state.

Zero dipolar energy can be achieved by canceling magnetic charges. Absence ofsurface charges M.n requires that magnetization remains parallel to the edge of thenanostructure (Figure II.10); this is a boundary condition. At any point P at theborder, let us consider the cartesian coordinates (x, y) with x and y respectivelytangent and inward normal to the boundary. The density of volume charges reads

II.4. Domains and domain walls in nanostructures 55

∂xmx + ∂ymy. As m lies along x, ∂xmx = 0. Thus cancelation of volume chargesis achieved if ∂ymy = 0; this is the differential equation to be solved. As my = 0at the boundary, absence of volume charges is fulfilled by keeping m normal to theradius originating from P .

Figure II.10: The principlefor building a magnetizaitonconfiguration free of dipolarfields.

Radii originating from different points at theboundary may intersect, each propagating inwardsmagnetization with a different direction, in whichcase highlighting the locus of a domain wall. It canbe demonstrated that domain walls in the nanos-tructure are at the loci of the centers of all circlesinscribed inside the boundary at two or more points.This geometrical construction satisfies that any do-main wall is bisecting the direction of magnetizationin the neighboring domains, a requirement pointedout in sec.3.2. Figure II.11a-b shows examples of theVan den Berg’s construction. A mechanical anal-ogy of this construction is sand piles, where lines ofequal height stand for flux lines.

Divide a nanostructure in two or more parts, apply the construction to each of thembefore bringing all parts back together: a higher order ground state is found with zerodipolar energy. An infinity of such states exists. In experiments such states may beprepared through special (de)magnetization procedures. High order states may also notbe stable in a real sample, because the wall width and energy neglected in the modelwill become prohibitively large. Notice also that the construction may still be used inthe case of a weak in-plane magnetic anisotropy in the sample, however suitably dividingthe sample in several parts with lines parallel to the easy axis of magnetization (Fig-ure II.11)c.

4.2 Domains in nanostructures with out-of-plane magneti-zation

Although to a lesser extent than for in-plane magnetization, domains of perpendicularly-magnetized material are influence by lateral finite-size effects. This is obviously thecase for weak-stripe domains, as a significant part of magnetization lies in-the-plane,calling for effects similar to those highlighted in the previous paragraph. Strongstripe domains may also be influenced in a flat nanostructure. Two arguments maybe put forward: the local demagnetizing field is smaller close to an edge, with re-spect to the core of a nanostructure; this would favor uniform magnetization closeto an edge, and thus local alignement of the stripes along this edge. Another argu-ment is that a stripe with opposite magnetization is ’missing’ beyond the border,removing a stabilizing effect on the stripe at the border; this would call for orientingstripes perpendicular to the border to better compensate surface charges. It seemsthat in some experiments the stripes display a tendency to align either parallel orperpendicular to the border, in the same sample[53]. For thick films it seems thatalignement of the stripes parallel to the border is favored[54].


Figure II.11: The geometrical construction of Van den Berg. (a) first order con-struction, along with a sand pile analogue (b) higher-order construction, along witha sand pile analogue (c) Kerr microscopy of an experimental realization of a highorder pattern from a stripe with an in-plane axis of anisotropy (sample courtesy:B. Viala, CEA-LETI).

4.3 The critical single-domain size

In the above we considered domains in large samples. We now examine down towhich size domains may be expected in nanostructures, called the critical single-domain size.

Let us consider a rather compact nanostructure, i.e. with all three demagnetizingcoefficient N close to 1/3, and lateral size l. If uniformly magnetized, its total energyis ESD = NKd

43πR3. We now have to discuss separately the cases of hard versus

soft magnetic materials.In hard magnetic materials domain walls are narrow and with an areal energy

density γW determined from materials properties. If split in two domains to closeits magnetic flux, the energy of such a nanostructure is ED ≈ εdNKd + l2γW withεD expressing the residual dipolar energy remaining despite the flux closure. γW =4√AKu in the case of uniaxial anisotropy. Equating ESD and ED yields the critical

single-domain size lSD = γW/[N(1 − εd)Kd] below which the single-domain stateis expected, while above which splitting into two or more domains is expected.lSD ≈ γW/NKd ≈

√AKu/Kd. lSD is of the order of one hundred nanometers for

permanent magnet materials.In soft magnetic materials a flux-closure state often takes the form of a collective

magnetization distribution, implying a slow rotation of magnetization as seen in Van

II.4. Domains and domain walls in nanostructures 57

(a) (b)

Figure II.12: Magnetization states of a disk of permalloy with diameter 100 nm andthickness 10 nm. The background color codes the y component of magnetization.Arrows stand for the magnetization vector. (a) near single-domain and (b) vortexstates.

den Berg’s constructions (sec.4.1). The relevant quantities are then exchange anddipolar energy, so that the critical single-domain size is expected to scale with thedipolar exchange length ∆d. Numerical simulation provides the numerical factor,lSD ≈ 7∆d for cubes and lSD ≈ 4∆d for spheres[6, p.156].

Estimating the critical single domain dimensions for non-compact nanostructures(i.e. with lengths quite different along the three directions) requires specific models.An important case is the transition from single-domain to the vortex state in a diskof diameter w and thickness t (Figure II.12). ESD ≈ NKdtw

2 with N ≈ t/w thein-plane demagnetizing coefficient. As a crude estimate the (lower bound for the)energy ED of the flux-closure state is the exchange plus dipolar energy of the core,round 10∆2

dtKd. Equating both we find the scaling law wt ≈ 10∆2d for the critical

dimensions. Numerical simulation provides an excellent agreement with the scalinglaw, however refines the numerics: wt ≈ 20∆2

d[55].

4.4 Near-single-domain

In the previous paragraph we discussed the scaling laws for dimensions, below whicha nanostructure does not display domains. Here we notice that such nanostructuresare often not perfectly uniformly-magnetized. We discuss the origins and the con-sequences of this effect.

When deriving the theory of demagnetization coefficients in sec.4.3, we noticedthat the self-consistence of the hypothesis of uniform magnetization may be satisfiedonly in the case of homogenous internal field. In turn, this may be achieved onlyin ellipsoids, infinite cylinders with elliptical cross-section, and slabs with infinitelateral dimensions. Many samples do not display such shapes, in particular flatstructures made by combining deposition and lithography. Figure II.13 shows thedemagnetizing field in a flat stripe assumed magnetized uniformly across its width.The field is highly non-homogeneous, being very intense close to the edges (mathe-matically, going towards Ms/2, and very weak in the center, below its average value


−NMsII.5. This is a practical example of the statement found in sec.2.1, about the

short range of dipolar fields for a two-dimensional nanostructure.

Figure II.13: Demagnetizing field in astripe magnetized uniformly across thewidth, of width 200 nm and thickness2.5 nm

Due to the high value of demag-netizing field close to the edges, mag-netization undergoes a strong torqueand cannot remain uniformly magne-tized, at least in the absence of anexternal field. The resulting areasare called end domains, with a ten-dency of magnetization to turn paral-lel to the edge to reduce edges chargesand instead spread them in the vol-ume. Although no real domains de-velop, this is a reminiscence of the Vanden Berg construction. In the case ofelongated elements, so-called ’S’ and’C’ states arise, named after the shapeof the flux lines, and reflecting themostly independence of end domainwhen sufficiently apart one from an-other (Figure II.14a-b).

Non-uniform magnetization configurations may persist down to very small size,especially close to corners where demagnetizing fields diverge in the mathematicallimit[56, 57]. This leads to the phenomenon of configurational anisotropy, describedboth analytically and computationnally[58, 59, 60]: certain directions for the averagemoment have an energy lower than others, arising from the orientation-dependantdecrease of dipolar energy (at the expense of exchange) made possible by the non-uniformity of magnetization. This effect adds up to the quadratic demagnetizingtensor, and may display symmetries forbidden by the latter, in relation with theshape of the element: order 3, 4, 5 etc (Figure II.14c-d). In sec.4.1 we will refer toa method to evaluate experimentally the strength of this anisotropy.

4.5 Domain walls in stripes and wires

We consider nanostructures elongated in one direction, which we will call wireswhen the sample dimensions are similar along the other two directions, and stripeswhen one of them is much smaller than the other. The latter is the case for mostsamples made by lithography, while the former is the case for samples made e.g.by electrodeposition in cylindrical pores[61]. We restrict the discussion to thosestripes and wires where no magnetocrystalline anisotropy is present, so that shapeanisotropy forces magnetization to lie along the axis. Domain walls may be foundin long objects, called head-to-head or tail-to-tail depending on the orientation ofmagnetization in the two segments.

Micromagnetic simulation predicts the existence of two main types of domainwalls for stripes: either the vortex wall (VW) or the transverse wall (TW) (Fig-ure II.15). The lowest energy is for the latter for tw < 61∆2

d, while the vortexdomain wall prevails at large thickness or width. Although this scaling law is simi-

II.5The analytical derivation of which is proposed in a problem

II.5. An overview of characteristic quantities 59

Figure II.14: Near-single-domain state in rectangles of dimensions 200×100×10 nmand squares of dimensions 100×100×10 nm. (a) S state (b) C state (c) flower state(d) leaf state.

lar to that of the single-domain-versus-vortex phase diagram for disks however witha larger coefficient (sec.4.3), its ground is slightly different. It was indeed noticedthat most of the energy in both the VW and TV are of dipolar origin[62], resultingfrom charges of the head-to-head or tail-to-tail. These charges are spread over theentire volume of the domain wall. Using integration of H2

d over space to estimatedipolar energy, and noticing that the surface of the TW is roughly twice as large asthat of the VW and the decay with height of Hd is roughly w, the tw scaling lawis again derived. Although both transverse and vortex domain walls are observedexperimentally, the range of metastability is large so that it is not possible to derivean experimental energetic phase diagram. TW may for instance be prepared far inthe metastability area through preparation with a magnetic field transverse to thestripe. For the largest thickness and especially width TW turn asymmetric (ATW)through a second-order transition.

5 An overview of characteristic quantities

In the course of this chapter we met many characteristic quantities: lengths, energies,dimensionless ratios etc. Here we make a short summary of them.


Figure II.15: Head-to-head domain walls in stripes, of (a) transverse and (b) vortextype.

5.1 Energy scales

• Kd = (1/2)µ0M2s is called the dipolar constant. It is a measure of the maxi-

mum dnsity of dipolar energy that can arise in a volume, i.e. for demagnetizingcoefficient N = 1.

• 4√AKu is the energy of a Bloch wall per unit area.

5.2 Length scales

• In a situation where only magnetic exchange and anisotropy compete, the tworelevant quantities in energy are A and Ku, expressed respectively in J/m andJ/m3. The typical case is that of a Bloch domain wall (sec.5). The resultinglength scale is ∆u =

√A/Ku. We call ∆u the anisotropy exchange length[10]

or Bloch parameter, a name often found in the literature. The latter is moreoften used, however the former makes more sense, see the note below. Noticethat ∆u is sometimes called the Bloch wall width, which however brings someconfusion as several definitions may be used for this, see sec.6.5.c.

• When exchange and dipolar energy compete, the two quantities at play areA and Kd. This the case in the vortex (sec.3.4). The resulting length scaleis ∆d =

√A/Kd =

√2A/µ0M2

s , which we call dipolar exchange length[6]or exchange length as more often found in the literature, see again the notebelow.

• lSD ≈√AKu/Kd is the critical domain size of a compact nanostructure made

of a quite hard magnetic material. It emerges out of the comparison of twoenergies, one per unit volume, the other one per unit surface.

• In more complex situations other length scales may arise, taking into accountan applied magnetic field, dimensionless quantities such as the ratio of geo-metric features etc. For example the pinning of a domain wall on a defectgives rise to the length scale

√A/µ0MsH for a soft magnetic material, or√

2A/√Kuµ0MsH for a material with significant magnetic anisotropy.

II.5. An overview of characteristic quantities 61

The name exchange length has historical grounds however is not well suited. Indeedexchange plays an equal role in both ∆u and ∆d. It is more relevant to name ∆u theanisotropy length or anisotropy exchange length, and ∆d the dipolar length or dipolarexchange length. We use the subscripts u (for uniaxial) and d (for dipolar) to accountfor this, as suggested in Hubert’s book[6].

5.3 Dimensionless ratios

• A quantity of interest in the quality factor Q = Ku/Kd, which describesthe competition between uniaxial anisotropy and dipolar energy. Q largelydetermines the occurrence and type of domains in thin films with an out-of-plane magnetocrystalline anisotropy.

Problems for Chapter II

Problem 1: Short exercises

1. Consider a cubic material with first-order magnetocrystalline anisotropy con-stant K1,cub much weaker than Kd, in the form of a thin film with surfacenormal (001).

• Express the resulting in-plane magnetic anisotropy E(θ) with θ the in-plane angle of magnetization with an easy axis, assuming that magneti-zation lies purely in-the-plane. Comment.

• Find exactly the easy directions of magnetization.

For both items consider both cases of positive and negative K1,cub, and com-ment.

2. Draw a sketch of the expected contrast in the magnetic microscopy of domainwalls. Consider four types of domain walls: perpendicular anisotropy withBloch wall; in-plane anisotropy with Bloch wall and Neel caps, 180 Neel walland 180 Neel wall. Consider four techniques: XMCD-PEEM, Lorentz, MFM,polar Kerr. The sketches may be presented as an array for clarity.

3. Derive with simple arguments the scaling law W ∼ t1/2 for the period of strongstripe domains.

Problem 2: Demagnetizing field in a stripe

Here we derive the analytical formula for the in-plane demagnetizing field ina flat and infinitely-long stripe magnetized in-the-plane, a case that was shortlydiscussed in sec.4.4. We call t and w its thickness and width, respectively. Weassume magnetization to be homogeneous and along the transverse direction.

2.1. Deriving the field

Express the stray field Hd arising from a line holding the magnetic charge perunit length λ. As a first step, we consider only Hd,x(x), the x component of thedemagnetizing field calculated at mid-height of the stripe, arising from the charges

62

Problem 2: Demagnetizing field in a stripe 63

on one of its edges. Write an integral form for this function. Show that it reads,upon integration:

x

z

θ

M

Figure II.16: Left part of the stripeconsidered. The edge holding themagnetic charges is highlighted as abold line. Magnetization is along x,while a translation invariance is as-sumed along y.

Hd,x(x) =Ms

2

[1− 2

πarctan

(2x

t

)](II.11)

2.2. Numerical evaluation andplotting

Derive the limits and first derivative forEq. (II.11) for x→ 0 and x→∞, and com-ment. Provide a hand-drawn qualitativeplot of this function. Without performingmore calculation, discuss how it comparesin magnitude with the z average over thethickness, i.e. 〈Hd,x,z〉 (x)? What is the(x, z) average of the latter over the entirecross-section of the stripe?

Chapter III

Magnetization reversal

1 Coherent rotation of magnetization

Overview

The importance of metastability in magnetism was outlined in 1.3, asthe reason for hysteresis. The determination of energy minima, landscapeand energy barriers is therefore crucial, however difficult or impossible inextended systems due to the large number of degrees of freedom. Onlysimple problems can be tackled analytically. Coherent rotation of mag-netization is one of the oldest and probably the most useful startingpoint.

1.1 The Stoner-Wohlfarth model

The model of coherent rotation was proposed by Stoner and Wohlfarth in 1948 todescribe the two-dimensional angular dependance of magnetization reversal[63, 64,reprint], and developed in parallel by Neel to describe thermally-activated processes.Many developments were made later, including clever graphical interpretations[65]andgeneralization to three dimensions[66].

The model is based on the hypothesis of uniform magnetization, reducing theproblem to solely one or two angular degrees of freedom. This hypothesis is inprinciple very restrictive and would be satisfactorily applicable to closely single-domain particles. For large systems it is not suitable as it, with for example anexperimental coercivity much smaller than the one predicted. Nevertheless, theconcept introduced for uniform magnetization bear some generality (e.g. exponents,angular dependance), and may be applied to extended systems with some care, e.g.to describe nucleation volumes.

We consider a system with volume V , total uniaxial anisotropy energy κ = KuV ,magnetic moment M = MsV . Its magnetic energy reads:

E = κ sin2 θ − µ0MH cos(θ − θH) (III.1)

where θH is the angle between the applied field and the easy axis and initial directionof magnetization, and H is positive to promote magnetization reversal. Here weconsider only the usual case of θH = π, and use dimensionless variable:

e = E/κ = sin2 θ + 2h cos θ (III.2)

64

III.2. Magnetization reversal in nanostructures 65

with Ha = 2K/µ0Ms and h = H/Ha. The equilibrium positions are determined bysolving dθe = 2 sin θ(cos θ − h) = 0, and stability with the sign of d2

e2θ = 4 cos2 θ −2h cos θ − 2.

Pour h < 1 les positions d’equilibre sont donc θ+1 = 0, θ−1 = π et θ2 tel que

cos θ2 = h. Pour ces angles la derivee seconde vaut 2(1− h), 2(1 + h) et 2(1− h2),respectivement. θ±1 sont donc des positions d’equilibre stables, et θ±2 = ± arccos(h)est une position d’equilibre instable, qui marque le sommet de la barriere d’energiequi empeche le renversement d’aimantation de θ+

1 vers θ−1 . Pour h > 1 il n’existeplus que θ−1 et θ+

1 comme points d’equilibre, respectivement stable et instable. Leretournement d’aimantation a donc lieu a h = 1.

1.2 Dynamic coercivity and temperature effects

2 Magnetization reversal in nanostructures

2.1 Multidomains under field (soft materials)

2.2 Nearly single domains

2.3 Domain walls and vortices

3 Magnetization reversal in extended systems

3.1 Nucleation and propagation

3.2 Ensembles of grains

Some features of the magnetization reversal of isolated single-domain grains havebeen presented in sec.1. Some consequences may be drawn for media consisting ofassemblies of grains, neglecting inter-grain interactions (dipolar etc). Of easy accessand modeling are the remanence mr and the internal energy at saturation EK ,derived from the area above the remagnetizing curve (i.e., starting from remanence,see sec.chap.I.1.3 and later sec.4.1). Both depend on the dimensionality of thedistribution of easy axis. Assuming uniaxial magnetic anisotropy for simplicity, weconsider three common cases:

• The polycrystalline case, i.e. with an isotropic distribution of easy axis inspace. This may correspond to particles diluted in a matrix, or a polycristallinebulk material. We then find: m3D

r = 1/2 and E3DK = 2K/3.

• The polytextured case. By this we mean a shared axis with no distributionfor the hard axis, while the easy axis if evenly distributed in the plane per-pendicular to this axis. This would be the case of Fe(110) grains grown on asurface, the easy axis lying along the in-plane [001] direction. When the fieldis applied in the plane we find: m2D

r = 2/π ≈ 0.64 and E2DK = K/2.

• The textured case, where all grains share the same direction of easy axis.When the field is applied along this axis we find the case of a single grain:m1D

r = 1 and E1DK = 0.

66 Chapter III. Magnetization reversal

The measure of EK provides an indication of K. Besides, comparison of experi-ments with the expected figure for mr is often used as an indication for interactions,positive (e.g. through direct exchange between neighboring grains) if the experimen-tal value exceeds the expectation, negative it it lies below. Systems with coupledgrains will be considered in more detail in chap.V.

4 What do we learn from hysteresis loops?

4.1 Magnetic anisotropy

Measure total anisotropy. Notice/reminder: area above curve. Linear for uniaxial,other shapes with possibly hysteresis, Cf (110), (001). Trick for these: loops undertransverse field. Show angular curves for anisotropy.

4.2 Nucleation versus propagation

First magnetization curves: nucleation versus propagationHc(theta), Kondorski

4.3 Distribution and interactions

Minor loops: reversible versus irreversible; hints for interactions.Interactions: Henkel plots.Complex: Preisach, now often called FORC.

Problems for Chapter III

Problem 1: Short exercises

1. Give a realistic example of a magnetically-uniaxial system whose coercivity Hc

is larger than its anisotropy field Ha.

2. Derive the formulas for remanence mr and remagnetization energy EK for thevarious cases of texture provided in sec.3.2.

Problem 2: A model of pinning - Kondorski’s law

for coercivity

We consider a one-dimensional framework, identical to the one used to derive theprofile of the Bloch domain wall, see pb. 6.5.c. Starting from a homogeneous materiallet us model a local defect in the form of a magnetically softer (i.e. anisotropyconstant K−∆K with ∆K > 0) insertion of width δ`, located at position x. Discusswhat approach should be followed to derive exactly the profile of the domain wall inthat case, especially the boundary conditions at the edges of the defect. To handlesimple algebra we make the assumption of the rigid domain wall, i.e. Eq. (I.37) stillholds, and consider the case where δ` ∆.

Show that the energy of the domain wall with the defect at location x reads:

E(x) = 4√AK

[1− 1

4

δ`

∆

δK

K

1

cosh2(x/∆)

](III.3)

Draw a schematic graph of E(x) and display the characteristic length or energyscales. An external field is then applied at an angle cos θH with the easy axis directionin the domains. Assuming that the assumption of rigid wall remains valid, show thatthe propagation field of the domain wall over the defect reads:

Hp =Ha

cos θH

∆K

K

δ`

∆

1

3√

3. (III.4)

where Ha = 2K/µ0Ms is the so-called anisotropy field.Notice:

67

68 Problems for Chapter III

• The 1/ cos θH dependence of coercivity is often considered as a signature aweak-pinning mechanism, a law known as the Kondorski model[67].

• This model had been initially published in 1939 by Becker and Doring[68],and is summarized e.g. in the nice book of Skomsky Simple models of Mag-netism[5].

• While coercivity requires a high anisotropy, the latter is not a sufficient condi-tion to have a high coercivity. To achieve this one must prevent magnetizationreversal that can be initiated on defects (structural or geometric) and switchthe entire magnetization by propagation of a domain wall. In a short-hand clas-sification one distinguishes coercivity made possible by hindering nucleation,or hindering the propagation of domain walls. In reality both phenomena areoften intermixed. Here we modeled an example of pinning.

• Simple micromagnetic models of nucleation on defects[69] were the first to beexhibited to tentatively explain the so-called Brown paradox, i.e. the fact thatvalues of experimental values of coercivity in most samples are smaller or muchsmaller than the values predicted by the ideal model of coherent rotation[63].

Chapter IV

Precessional dynamics ofmagnetization

1 Ferromagnetic resonance and Landau-Lifshitz-

Gilbert equation

2 Precessional switching of macrospins driven by

magnetic fields

3 Precessional switching driven by spin transfer

torques

4 Precessional dynamics of domain walls and vor-

tices – Field and current

69

Problems for Chapter IV

70

Chapter V

Magnetic heterostructures: fromspecific properties to applications

1 Coupling effects

2 Magnetotransport

3 Integration for applications

71

Problems for Chapter V

72

Appendices

Symbols

Kd Dipolar anisotropy Ks = 12µ0M

2s

Q Quality factor Q = Kmc/Kd

∆u Anisotropy exchange length ∆ =√A/K with A the exchange and K the

anisotropy constant. Also called: Bloch wallparameter

∆d Dipolar exchange length Λ =√A/Kd =

√2A/µ0M2

s with A the ex-change and Ms the spontaneous magnetiza-tion. Also called: exchange length.

Acronyms

AFM Atomic Force MicroscopyEMF Electromotive forceMFM Magnetic Force MicroscopyPMA Perpendicular Magnetic AnisotropySEMPA Scanning Electron Microscopy with Polarization Analysis (SEMPA)SPLEEM Spin-Polarized Low-Energy Electron MicroscopySQUID Superconducting Quantum Interference DeviceUHV Ultra-High VacuumVSM Vibrating Sample Magnetometer

73

74 Appendices

Glossary

erg Unit for energy in the cgs-Gauss system. Is equivalent to 1010 J.Macrospin The model where uniform magnetization is assumed in a sys-

tem, whose description may thus be restricted to the knowledgeof one or two degrees of freedom, the angular directions of ahypothetical spin. When formerly written as a variable, themacrospin may be dimensionless, or have units of A.m2 for avolume, A.m for magnetization integrated over a surface (e.g.that of a nanowire), or A for magnetization integrated along athickness (e.g. that of a thin film).

Micromagnetism All aspects related the arrangement of magnetization in do-mains and domain walls, when the latter are resolved (i.e.,not treated as a plane with zero thickness nor energy). Theterm applies to theory, simulation and experiments. Exceptsome rare cases that may be considered as fine points, micro-magnetism is based on the description of magnetization by acontinuous function of constant and homogeneous magnitudeequal to the spontaneous magnetization Ms.

Nanomagnetism Broadly speaking, all aspects of magnetism at small lengthscale, typically below one micrometer. This concerns ground-state (intrinsic) properties such as magnetic ordering and mag-netic anisotropy, as well as magnetization configurations andmagnetization reversal at these small scales. Notice that somepersons restrict the meaning of Nanomagnetism to the former

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A faire...

Finalisation de poly ED

• F/AF: voir historique et references dans talk Bernard Dieny, livre conferencesI, p15 (CLN9).

• Citer pour coercitivite SyAF: H. A. M. van den Berg, W. Clemens, G. Gieres,G. Rupp, and M. Vieth, IEEE Trans. Magn. 32, 4624 (1996) (cite dans Wiese2004).

• courbe FC/ZFC: traiter le cas a une particule (phy stat et susceptibilitespremiere et seconde) puis le cas de distribution. Definition de Tb moyen dansce cas. Faire le lien avec distribution de champ de renversement, et definitionde Hc comme champ de mi-renversement.

• utilisation de hyperref pour les notes de bas de page?

• Noter superparamagnetisme de bandes etroites: LEE2009b.

•

• Nouvel environnement pour:

– Ameliorations

– Exercices et corriges

– Partie difficile (zigzag dans la marge, Cf bouquin Knuth, ou p.76 desymbols-a4.pdf, dbend)

– Verifier presence d’introductions et conclusion locales

– Renversement d’aimantation dans lignes:

∗ Hausmanns2002[70]: note que champ coercitif augmente comme 1/w,mais pas de modele.

∗ BRA2006: cite ceux precedents qui ont note cette loi d’echelle. In-dique egalement separation entre nucleation-propagation, et multido-maines (ligne tres differente de regime transverse-vortex).

∗ Un des premiers modeles: YUA1992

∗ Utiliser: Uhlig

– Interactions dipolaires: Cf etudes suppression superpara par Cowburn,et changement energie demagnetisante dans JAI2010.

• Regarder exemples presentations, et chercher code:

– These Geraud Moulas

– These Fabien Cheynis

– These Aurelien Massebœuf (Cf: p.62)

82 Bibliography

Generalites

• Tableau des materiaux: rajouter:

– Permalloy (et supermalloy)

– indiquer echange, longueurs echange et Bloch.

• Changer style de macro lecturePage

• Chercher si package derivees est defini quelque part.

• Regarder utilisation package vector

• Test page gauche ou droite pour mettre les encadres et leur image. (utilisermacro: ifodd)

• Redefinir un environnement enumerate pour les encadres, pour eviter espacevide en haut. Voir The tendency to cancel surface magnetic charges mais enenlevant aussi la marge.

• Changer police de la legende

• Mentionner temps tres raccourci entre developpement dans laboratoires, etmise en application dans l’industrie. Pour certains secteurs la RD est memeplus active que laboratoires fondamentaux, par exemple pour maıtriser lesprocessus de renversement d’aimantation dans les nanostructures.

• Employer Configuration magnetique ou Etat micromagnetique, et introduireles deux termes. Verifier dans tout le texte que pas de melange.

• Normaliser l’usage de termes suivants, pour les renvois:

– Chapitre

– Partie

– Paragraphe

• Mettre un glossaire, et un lien vers le glossaire a chaque fois que le terme estemploye (rechercher dans litterature).

• Verifier presence: differents modes de propagation des ondes de spin, voir coursHillebrands. Traitement classique et quantique des ODS (exercice?).

Exercices possibles

• Position et valeur du maximum pour le J avec regles de Hund, pour ` quel-conque. Meme chose pour valeur de moment magnetique.

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